Theorem resima2 5097

Description: Image under a restricted class. (Contributed by FL, 31-Aug-2009.)

Assertion

Ref	Expression
resima2	|- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) )

Proof of Theorem resima2

Step	Hyp	Ref	Expression
1		df-ima 4787	. 2 |- ( ( A |` C ) " B ) = ran ( ( A |` C ) |` B )
2		resres 5075	. . . 4 |- ( ( A |` C ) |` B ) = ( A |` ( C i^i B ) )
3	2	rneqi 5010	. . 3 |- ran ( ( A |` C ) |` B ) = ran ( A |` ( C i^i B ) )
4		df-ss 3233	. . . 4 |- ( B C_ C <-> ( B i^i C ) = B )
5		incom 3421	. . . . . . . 8 |- ( C i^i B ) = ( B i^i C )
6	5	a1i 9	. . . . . . 7 |- ( ( B i^i C ) = B -> ( C i^i B ) = ( B i^i C ) )
7	6	reseq2d 5063	. . . . . 6 |- ( ( B i^i C ) = B -> ( A |` ( C i^i B ) ) = ( A |` ( B i^i C ) ) )
8	7	rneqd 5011	. . . . 5 |- ( ( B i^i C ) = B -> ran ( A |` ( C i^i B ) ) = ran ( A |` ( B i^i C ) ) )
9		reseq2 5058	. . . . . . 7 |- ( ( B i^i C ) = B -> ( A |` ( B i^i C ) ) = ( A |` B ) )
10	9	rneqd 5011	. . . . . 6 |- ( ( B i^i C ) = B -> ran ( A |` ( B i^i C ) ) = ran ( A |` B ) )
11		df-ima 4787	. . . . . 6 |- ( A " B ) = ran ( A |` B )
12	10, 11	eqtr4di 2289	. . . . 5 |- ( ( B i^i C ) = B -> ran ( A |` ( B i^i C ) ) = ( A " B ) )
13	8, 12	eqtrd 2271	. . . 4 |- ( ( B i^i C ) = B -> ran ( A |` ( C i^i B ) ) = ( A " B ) )
14	4, 13	sylbi 121	. . 3 |- ( B C_ C -> ran ( A |` ( C i^i B ) ) = ( A " B ) )
15	3, 14	eqtrid 2283	. 2 |- ( B C_ C -> ran ( ( A |` C ) |` B ) = ( A " B ) )
16	1, 15	eqtrid 2283	1 |- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   i^i cin 3219   C_ wss 3220   ran crn 4775   |` cres 4776   "cima 4777

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787

This theorem is used by:  ressuppss  6494  cnptopresti  15339  cnptoprest  15340