Theorem resima2 5039

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 4732	. 2 |- ( ( A |` C ) " B ) = ran ( ( A |` C ) |` B )
2		resres 5017	. . . 4 |- ( ( A |` C ) |` B ) = ( A |` ( C i^i B ) )
3	2	rneqi 4952	. . 3 |- ran ( ( A |` C ) |` B ) = ran ( A |` ( C i^i B ) )
4		df-ss 3210	. . . 4 |- ( B C_ C <-> ( B i^i C ) = B )
5		incom 3396	. . . . . . . 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 5005	. . . . . 6 |- ( ( B i^i C ) = B -> ( A |` ( C i^i B ) ) = ( A |` ( B i^i C ) ) )
8	7	rneqd 4953	. . . . 5 |- ( ( B i^i C ) = B -> ran ( A |` ( C i^i B ) ) = ran ( A |` ( B i^i C ) ) )
9		reseq2 5000	. . . . . . 7 |- ( ( B i^i C ) = B -> ( A |` ( B i^i C ) ) = ( A |` B ) )
10	9	rneqd 4953	. . . . . 6 |- ( ( B i^i C ) = B -> ran ( A |` ( B i^i C ) ) = ran ( A |` B ) )
11		df-ima 4732	. . . . . 6 |- ( A " B ) = ran ( A |` B )
12	10, 11	eqtr4di 2280	. . . . 5 |- ( ( B i^i C ) = B -> ran ( A |` ( B i^i C ) ) = ( A " B ) )
13	8, 12	eqtrd 2262	. . . 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 2274	. 2 |- ( B C_ C -> ran ( ( A |` C ) |` B ) = ( A " B ) )
16	1, 15	eqtrid 2274	1 |- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) )

Syntax hints:   -> wi 4   = wceq 1395   i^i cin 3196   C_ wss 3197   ran crn 4720   |` cres 4721   "cima 4722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732

This theorem is referenced by:  cnptopresti  14912  cnptoprest  14913