Theorem cores2 5648

Description: Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)

Assertion

Ref	Expression
cores2	|- ( dom A C_ C -> ( A o. `' ( `' B |` C ) ) = ( A o. B ) )

Proof of Theorem cores2

Step	Hyp	Ref	Expression
1		dfdm4 5316	. . . . . 6 |- dom A = ran `' A
2	1	sseq1i 3629	. . . . 5 |- ( dom A C_ C <-> ran `' A C_ C )
3		cores 5638	. . . . 5 |- ( ran `' A C_ C -> ( ( `' B |` C ) o. `' A ) = ( `' B o. `' A ) )
4	2, 3	sylbi 207	. . . 4 |- ( dom A C_ C -> ( ( `' B |` C ) o. `' A ) = ( `' B o. `' A ) )
5		cnvco 5308	. . . . 5 |- `' ( A o. `' ( `' B |` C ) ) = ( `' `' ( `' B |` C ) o. `' A )
6		cocnvcnv1 5646	. . . . 5 |- ( `' `' ( `' B |` C ) o. `' A ) = ( ( `' B |` C ) o. `' A )
7	5, 6	eqtri 2644	. . . 4 |- `' ( A o. `' ( `' B |` C ) ) = ( ( `' B |` C ) o. `' A )
8		cnvco 5308	. . . 4 |- `' ( A o. B ) = ( `' B o. `' A )
9	4, 7, 8	3eqtr4g 2681	. . 3 |- ( dom A C_ C -> `' ( A o. `' ( `' B |` C ) ) = `' ( A o. B ) )
10	9	cnveqd 5298	. 2 |- ( dom A C_ C -> `' `' ( A o. `' ( `' B |` C ) ) = `' `' ( A o. B ) )
11		relco 5633	. . 3 |- Rel ( A o. `' ( `' B |` C ) )
12		dfrel2 5583	. . 3 |- ( Rel ( A o. `' ( `' B |` C ) ) <-> `' `' ( A o. `' ( `' B |` C ) ) = ( A o. `' ( `' B |` C ) ) )
13	11, 12	mpbi 220	. 2 |- `' `' ( A o. `' ( `' B |` C ) ) = ( A o. `' ( `' B |` C ) )
14		relco 5633	. . 3 |- Rel ( A o. B )
15		dfrel2 5583	. . 3 |- ( Rel ( A o. B ) <-> `' `' ( A o. B ) = ( A o. B ) )
16	14, 15	mpbi 220	. 2 |- `' `' ( A o. B ) = ( A o. B )
17	10, 13, 16	3eqtr3g 2679	1 |- ( dom A C_ C -> ( A o. `' ( `' B |` C ) ) = ( A o. B ) )

Syntax hints:   -> wi 4   = wceq 1483   C_ wss 3574   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   Rel wrel 5119

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  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-co 5123  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  fcoi1  6078  ofco2  20257