Theorem cores2 5194

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 4869	. . . . . 6 |- dom A = ran `' A
2	1	sseq1i 3218	. . . . 5 |- ( dom A C_ C <-> ran `' A C_ C )
3		cores 5185	. . . . 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 4862	. . . . 5 |- `' ( A o. `' ( `' B |` C ) ) = ( `' `' ( `' B |` C ) o. `' A )
6		cocnvcnv1 5192	. . . . 5 |- ( `' `' ( `' B |` C ) o. `' A ) = ( ( `' B |` C ) o. `' A )
7	5, 6	eqtri 2225	. . . 4 |- `' ( A o. `' ( `' B |` C ) ) = ( ( `' B |` C ) o. `' A )
8		cnvco 4862	. . . 4 |- `' ( A o. B ) = ( `' B o. `' A )
9	4, 7, 8	3eqtr4g 2262	. . 3 |- ( dom A C_ C -> `' ( A o. `' ( `' B |` C ) ) = `' ( A o. B ) )
10	9	cnveqd 4853	. 2 |- ( dom A C_ C -> `' `' ( A o. `' ( `' B |` C ) ) = `' `' ( A o. B ) )
11		relco 5180	. . 3 |- Rel ( A o. `' ( `' B |` C ) )
12		dfrel2 5132	. . 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 5180	. . 3 |- Rel ( A o. B )
15		dfrel2 5132	. . 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 2260	1 |- ( dom A C_ C -> ( A o. `' ( `' B |` C ) ) = ( A o. B ) )

Syntax hints:   -> wi 4   = wceq 1372   C_ wss 3165   `'ccnv 4673   dom cdm 4674   ran crn 4675   |` cres 4676   o. ccom 4678   Rel wrel 4679

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686

This theorem is referenced by:  cocnvres  5206  fcoi1  5455