Theorem resima2 4901

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 4600	. 2 |- ( ( A |` C ) " B ) = ran ( ( A |` C ) |` B )
2		resres 4879	. . . 4 |- ( ( A |` C ) |` B ) = ( A |` ( C i^i B ) )
3	2	rneqi 4815	. . 3 |- ran ( ( A |` C ) |` B ) = ran ( A |` ( C i^i B ) )
4		df-ss 3115	. . . 4 |- ( B C_ C <-> ( B i^i C ) = B )
5		incom 3299	. . . . . . . 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 4867	. . . . . 6 |- ( ( B i^i C ) = B -> ( A |` ( C i^i B ) ) = ( A |` ( B i^i C ) ) )
8	7	rneqd 4816	. . . . 5 |- ( ( B i^i C ) = B -> ran ( A |` ( C i^i B ) ) = ran ( A |` ( B i^i C ) ) )
9		reseq2 4862	. . . . . . 7 |- ( ( B i^i C ) = B -> ( A |` ( B i^i C ) ) = ( A |` B ) )
10	9	rneqd 4816	. . . . . 6 |- ( ( B i^i C ) = B -> ran ( A |` ( B i^i C ) ) = ran ( A |` B ) )
11		df-ima 4600	. . . . . 6 |- ( A " B ) = ran ( A |` B )
12	10, 11	eqtr4di 2208	. . . . 5 |- ( ( B i^i C ) = B -> ran ( A |` ( B i^i C ) ) = ( A " B ) )
13	8, 12	eqtrd 2190	. . . 4 |- ( ( B i^i C ) = B -> ran ( A |` ( C i^i B ) ) = ( A " B ) )
14	4, 13	sylbi 120	. . 3 |- ( B C_ C -> ran ( A |` ( C i^i B ) ) = ( A " B ) )
15	3, 14	syl5eq 2202	. 2 |- ( B C_ C -> ran ( ( A |` C ) |` B ) = ( A " B ) )
16	1, 15	syl5eq 2202	1 |- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) )

Syntax hints:   -> wi 4   = wceq 1335   i^i cin 3101   C_ wss 3102   ran crn 4588   |` cres 4589   "cima 4590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-xp 4593  df-rel 4594  df-cnv 4595  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600

This theorem is referenced by:  cnptopresti  12680  cnptoprest  12681