Theorem resima2OLD 5433

Description: Obsolete proof of resima2 5432 as of 25-Aug-2021. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
resima2OLD	|- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) )

Proof of Theorem resima2OLD

Step	Hyp	Ref	Expression
1		df-ima 5127	. 2 |- ( ( A |` C ) " B ) = ran ( ( A |` C ) |` B )
2		resres 5409	. . . 4 |- ( ( A |` C ) |` B ) = ( A |` ( C i^i B ) )
3	2	rneqi 5352	. . 3 |- ran ( ( A |` C ) |` B ) = ran ( A |` ( C i^i B ) )
4		df-ss 3588	. . . 4 |- ( B C_ C <-> ( B i^i C ) = B )
5		incom 3805	. . . . . . . 8 |- ( C i^i B ) = ( B i^i C )
6	5	a1i 11	. . . . . . 7 |- ( ( B i^i C ) = B -> ( C i^i B ) = ( B i^i C ) )
7	6	reseq2d 5396	. . . . . 6 |- ( ( B i^i C ) = B -> ( A |` ( C i^i B ) ) = ( A |` ( B i^i C ) ) )
8	7	rneqd 5353	. . . . 5 |- ( ( B i^i C ) = B -> ran ( A |` ( C i^i B ) ) = ran ( A |` ( B i^i C ) ) )
9		reseq2 5391	. . . . . . 7 |- ( ( B i^i C ) = B -> ( A |` ( B i^i C ) ) = ( A |` B ) )
10	9	rneqd 5353	. . . . . 6 |- ( ( B i^i C ) = B -> ran ( A |` ( B i^i C ) ) = ran ( A |` B ) )
11		df-ima 5127	. . . . . 6 |- ( A " B ) = ran ( A |` B )
12	10, 11	syl6eqr 2674	. . . . 5 |- ( ( B i^i C ) = B -> ran ( A |` ( B i^i C ) ) = ( A " B ) )
13	8, 12	eqtrd 2656	. . . 4 |- ( ( B i^i C ) = B -> ran ( A |` ( C i^i B ) ) = ( A " B ) )
14	4, 13	sylbi 207	. . 3 |- ( B C_ C -> ran ( A |` ( C i^i B ) ) = ( A " B ) )
15	3, 14	syl5eq 2668	. 2 |- ( B C_ C -> ran ( ( A |` C ) |` B ) = ( A " B ) )
16	1, 15	syl5eq 2668	1 |- ( B C_ C -> ( ( A |` C ) " B ) = ( A " B ) )

Syntax hints:   -> wi 4   = wceq 1483   i^i cin 3573   C_ wss 3574   ran crn 5115   |` cres 5116   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by: (None)