Theorem resima2 4848

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 4547	. 2 |- ( ( A |` C ) " B ) = ran ( ( A |` C ) |` B )
2		resres 4826	. . . 4 |- ( ( A |` C ) |` B ) = ( A |` ( C i^i B ) )
3	2	rneqi 4762	. . 3 |- ran ( ( A |` C ) |` B ) = ran ( A |` ( C i^i B ) )
4		df-ss 3079	. . . 4 |- ( B C_ C <-> ( B i^i C ) = B )
5		incom 3263	. . . . . . . 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 4814	. . . . . 6 |- ( ( B i^i C ) = B -> ( A |` ( C i^i B ) ) = ( A |` ( B i^i C ) ) )
8	7	rneqd 4763	. . . . 5 |- ( ( B i^i C ) = B -> ran ( A |` ( C i^i B ) ) = ran ( A |` ( B i^i C ) ) )
9		reseq2 4809	. . . . . . 7 |- ( ( B i^i C ) = B -> ( A |` ( B i^i C ) ) = ( A |` B ) )
10	9	rneqd 4763	. . . . . 6 |- ( ( B i^i C ) = B -> ran ( A |` ( B i^i C ) ) = ran ( A |` B ) )
11		df-ima 4547	. . . . . 6 |- ( A " B ) = ran ( A |` B )
12	10, 11	syl6eqr 2188	. . . . 5 |- ( ( B i^i C ) = B -> ran ( A |` ( B i^i C ) ) = ( A " B ) )
13	8, 12	eqtrd 2170	. . . 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 2182	. 2 |- ( B C_ C -> ran ( ( A |` C ) |` B ) = ( A " B ) )
16	1, 15	syl5eq 2182	1 |- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) )

Syntax hints:   -> wi 4   = wceq 1331   i^i cin 3065   C_ wss 3066   ran crn 4535   |` cres 4536   "cima 4537

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547

This theorem is referenced by:  cnptopresti  12396  cnptoprest  12397