BAGHDAD -- Iraq’s air force struck ISIS targets inside Syria for the first time on Friday as Iraqi troops on the ground pushed into western Mosul, the last major urban stronghold held by the Sunni militant group in Iraq.

Iraq’s Prime Minister Haider al-Abadi announced the airstrikes in a statement, saying the air force hit towns of Boukamal and Husseibah across the border and came in response to recent bombings in Baghdad claimed by ISIS and linked to the militants’ operations in Syria.

Meanwhile, Iraqi forces pushed into the first neighborhood of western Mosul and took full control of Mosul’s international airport and a sprawling military base on the southwestern edge of the city, according to Iraqi officials.

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The territorial gains mark the first key moves in the battle, now in its sixth day, to rout ISIS militants from the western half of Mosul, Iraq’s second-largest city.

CBS News’ David Martin has been traveling with the commander of U.S. forces in the Middle East to get a rare, up-close look at how American forces are assisting in the fight against ISIS in both Iraq and Syria.

The liberation of Mosul, along with the ISIS’ de-facto capital of Raqqa in neighboring Syria, would be impossible without the airstrikes flown by American and allied jets from bases dotted across the region. At one of those bases, Gen. Joseph Votel told the U.S. troops that despite all the U.S., assistance, it is up to the Iraqis to win on the ground in Mosul.

“They own this. We are helping them to fight but we are not doing the fighting for them,” he said.

To increase the pressure on ISIS, the U.S. has now moved some precision guided rocket launchers, weapons known as HIMARs, to a position inside Syria itself.

Martin says the U.S. currently has about 500 troops inside Syria, but the Pentagon is considering adding hundreds more to assist in the liberation of Raqqa.

The push into Mamun neighborhood in western Mosul was followed by intense clashes with ISIS militants, according to an Iraqi special forces officer on the ground, who spoke on condition of anonymity in line with regulations.

An Associated Press team near the front line saw at least four wounded special forces’ members and the bodies of three soldiers, suggesting more intense fighting than the previous day. Iraq’s military does not release official casualty information.

Earlier on Friday, the spokesman of the Joint Military Operation Command, Brig. Gen. Yahya Rasool, said Iraqi forces had also retaken the military base adjacent to the airport.

An Iraqi air force commander said the air strikes against ISIS in Syria carried out with F-16 warplanes at dawn and “were successful.” The commander, who spoke on condition of anonymity because he was not authorized to talk to the media, said they were conducted at the order of the prime minister.

The advances come a day after special forces joined the fight for western Mosul.

Both the Ghazlani military base and the airport will be key to the next steps in the daunting battle and will serve as a base of operations as Iraqi forces launch subsequent pushes into western Mosul, which is divided by the Tigris River into two halves.

Iraqi authorities declared the city’s eastern half “fully liberated” from the Sunni militants in January, three months after launching the operation to take back Mosul, Iraq’s second-largest city.

On Thursday, Iraqi special forces joined federal police and rapid response units in the push while the Popular Mobilization Forces -- an umbrella group of government-sanctioned Shiite militias -- secured the main roads west of Mosul, largely cutting the city off from ISIS-held territory in Syria.

The United Nations estimated that about 750,000 civilians are trapped in western Mosul. The initial numbers of displaced from western Mosul have been low, but Iraqi forces are yet to punch into the city’s dense urban neighborhoods.

The battle for western Mosul is expected to be the most trying yet. The western half of the city is denser with older neighborhoods and narrower streets that will likely complicate the already difficult urban combat ahead.