Theorem cores2 5116

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 4796	. . . . . 6 |- dom A = ran `' A
2	1	sseq1i 3168	. . . . 5 |- ( dom A C_ C <-> ran `' A C_ C )
3		cores 5107	. . . . 5 |- ( ran `' A C_ C -> ( ( `' B |` C ) o. `' A ) = ( `' B o. `' A ) )
4	2, 3	sylbi 120	. . . 4 |- ( dom A C_ C -> ( ( `' B |` C ) o. `' A ) = ( `' B o. `' A ) )
5		cnvco 4789	. . . . 5 |- `' ( A o. `' ( `' B |` C ) ) = ( `' `' ( `' B |` C ) o. `' A )
6		cocnvcnv1 5114	. . . . 5 |- ( `' `' ( `' B |` C ) o. `' A ) = ( ( `' B |` C ) o. `' A )
7	5, 6	eqtri 2186	. . . 4 |- `' ( A o. `' ( `' B |` C ) ) = ( ( `' B |` C ) o. `' A )
8		cnvco 4789	. . . 4 |- `' ( A o. B ) = ( `' B o. `' A )
9	4, 7, 8	3eqtr4g 2224	. . 3 |- ( dom A C_ C -> `' ( A o. `' ( `' B |` C ) ) = `' ( A o. B ) )
10	9	cnveqd 4780	. 2 |- ( dom A C_ C -> `' `' ( A o. `' ( `' B |` C ) ) = `' `' ( A o. B ) )
11		relco 5102	. . 3 |- Rel ( A o. `' ( `' B |` C ) )
12		dfrel2 5054	. . 3 |- ( Rel ( A o. `' ( `' B |` C ) ) <-> `' `' ( A o. `' ( `' B |` C ) ) = ( A o. `' ( `' B |` C ) ) )
13	11, 12	mpbi 144	. 2 |- `' `' ( A o. `' ( `' B |` C ) ) = ( A o. `' ( `' B |` C ) )
14		relco 5102	. . 3 |- Rel ( A o. B )
15		dfrel2 5054	. . 3 |- ( Rel ( A o. B ) <-> `' `' ( A o. B ) = ( A o. B ) )
16	14, 15	mpbi 144	. 2 |- `' `' ( A o. B ) = ( A o. B )
17	10, 13, 16	3eqtr3g 2222	1 |- ( dom A C_ C -> ( A o. `' ( `' B |` C ) ) = ( A o. B ) )

Syntax hints:   -> wi 4   = wceq 1343   C_ wss 3116   `'ccnv 4603   dom cdm 4604   ran crn 4605   |` cres 4606   o. ccom 4608   Rel wrel 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616

This theorem is referenced by:  cocnvres  5128  fcoi1  5368