Rest batted after tea to reply with 36 for 1 in 16 overs on day two of the five-day match at the Brabourne Stadium.
The batsman out was opener K S Bharat, caught at point while cutting an Abhishek Nayar delivery. At stumps Faiz Fazal (18) and night watchman Jayant Yadav (1) were at the crease.
Earlier Yadav, unbeaten on 88 in last evening's Mumbai total of 386 for 3, went on to punish the toothless Rest bowling to make 156 in 271 balls, striking a six and 24 fours.
Yadav, who had scored a hundred in the Ranji Trophy semi-final against Madhya Pradesh at Cuttack, was finally dismissed by off spinner Jayant Yadav who accepted a return catch off an attempted drive. The same bowler then sent back Abhishek Nayar and Dhawal Kulkarni in successive overs.
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But by then Mumbai had already crossed the 500 mark and useful contributions from tail-enders Iqbal Abdullah (29) and Shardul Thakur (18) enabled them to cross the 600 mark.
Rest started their reply cautiously before losing Bharat and at stumps were trailing the Ranji champions by 567 runs.
Brief scores: Mumbai 603 all out (A Herwadkar 90, J Bista 104, S Iyer 55, S Yadav 156, A Tare 65, S Lad 66; J Unadkat 2 for 128, K Das 2 for 152, J Yadav 4 for 132) v Rest of India 36 for 1 (F Fazal 18 not out, J Yadav 1 not out).