Theorem cores2 5295

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 4968	. . . . . 6 |- dom A = ran `' A
2	1	sseq1i 3274	. . . . 5 |- ( dom A C_ C <-> ran `' A C_ C )
3		cores 5286	. . . . 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 4960	. . . . 5 |- `' ( A o. `' ( `' B |` C ) ) = ( `' `' ( `' B |` C ) o. `' A )
6		cocnvcnv1 5293	. . . . 5 |- ( `' `' ( `' B |` C ) o. `' A ) = ( ( `' B |` C ) o. `' A )
7	5, 6	eqtri 2259	. . . 4 |- `' ( A o. `' ( `' B |` C ) ) = ( ( `' B |` C ) o. `' A )
8		cnvco 4960	. . . 4 |- `' ( A o. B ) = ( `' B o. `' A )
9	4, 7, 8	3eqtr4g 2296	. . 3 |- ( dom A C_ C -> `' ( A o. `' ( `' B |` C ) ) = `' ( A o. B ) )
10	9	cnveqd 4951	. 2 |- ( dom A C_ C -> `' `' ( A o. `' ( `' B |` C ) ) = `' `' ( A o. B ) )
11		relco 5281	. . 3 |- Rel ( A o. `' ( `' B |` C ) )
12		dfrel2 5233	. . 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 5281	. . 3 |- Rel ( A o. B )
15		dfrel2 5233	. . 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 2294	1 |- ( dom A C_ C -> ( A o. `' ( `' B |` C ) ) = ( A o. B ) )

Syntax hints:   -> wi 4   = wceq 1402   C_ wss 3220   `'ccnv 4768   dom cdm 4769   ran crn 4770   |` cres 4771   o. ccom 4773   Rel wrel 4774

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781

This theorem is referenced by:  cocnvres  5307  fcoi1  5567