Theorem cores2 5256

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 4929	. . . . . 6 |- dom A = ran `' A
2	1	sseq1i 3254	. . . . 5 |- ( dom A C_ C <-> ran `' A C_ C )
3		cores 5247	. . . . 5 |- ( ran `' A C_ C -> ( ( `' B |` C ) o. `' A ) = ( `' B o. `' A ) )
4	2, 3	sylbi 121	. . . 4 |- ( dom A C_ C -> ( ( `' B |` C ) o. `' A ) = ( `' B o. `' A ) )
5		cnvco 4921	. . . . 5 |- `' ( A o. `' ( `' B |` C ) ) = ( `' `' ( `' B |` C ) o. `' A )
6		cocnvcnv1 5254	. . . . 5 |- ( `' `' ( `' B |` C ) o. `' A ) = ( ( `' B |` C ) o. `' A )
7	5, 6	eqtri 2252	. . . 4 |- `' ( A o. `' ( `' B |` C ) ) = ( ( `' B |` C ) o. `' A )
8		cnvco 4921	. . . 4 |- `' ( A o. B ) = ( `' B o. `' A )
9	4, 7, 8	3eqtr4g 2289	. . 3 |- ( dom A C_ C -> `' ( A o. `' ( `' B |` C ) ) = `' ( A o. B ) )
10	9	cnveqd 4912	. 2 |- ( dom A C_ C -> `' `' ( A o. `' ( `' B |` C ) ) = `' `' ( A o. B ) )
11		relco 5242	. . 3 |- Rel ( A o. `' ( `' B |` C ) )
12		dfrel2 5194	. . 3 |- ( Rel ( A o. `' ( `' B |` C ) ) <-> `' `' ( A o. `' ( `' B |` C ) ) = ( A o. `' ( `' B |` C ) ) )
13	11, 12	mpbi 145	. 2 |- `' `' ( A o. `' ( `' B |` C ) ) = ( A o. `' ( `' B |` C ) )
14		relco 5242	. . 3 |- Rel ( A o. B )
15		dfrel2 5194	. . 3 |- ( Rel ( A o. B ) <-> `' `' ( A o. B ) = ( A o. B ) )
16	14, 15	mpbi 145	. 2 |- `' `' ( A o. B ) = ( A o. B )
17	10, 13, 16	3eqtr3g 2287	1 |- ( dom A C_ C -> ( A o. `' ( `' B |` C ) ) = ( A o. B ) )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3201   `'ccnv 4730   dom cdm 4731   ran crn 4732   |` cres 4733   o. ccom 4735   Rel wrel 4736

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743

This theorem is referenced by:  cocnvres  5268  fcoi1  5525