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Theorem cores2 5382
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
StepHypRef Expression
1 dfdm4 5063 . . . . . 6  |-  dom  A  =  ran  `' A
21sseq1i 3372 . . . . 5  |-  ( dom 
A  C_  C  <->  ran  `' A  C_  C )
3 cores 5373 . . . . 5  |-  ( ran  `' A  C_  C  -> 
( ( `' B  |`  C )  o.  `' A )  =  ( `' B  o.  `' A ) )
42, 3sylbi 188 . . . 4  |-  ( dom 
A  C_  C  ->  ( ( `' B  |`  C )  o.  `' A )  =  ( `' B  o.  `' A ) )
5 cnvco 5056 . . . . 5  |-  `' ( A  o.  `' ( `' B  |`  C ) )  =  ( `' `' ( `' B  |`  C )  o.  `' A )
6 cocnvcnv1 5380 . . . . 5  |-  ( `' `' ( `' B  |`  C )  o.  `' A )  =  ( ( `' B  |`  C )  o.  `' A )
75, 6eqtri 2456 . . . 4  |-  `' ( A  o.  `' ( `' B  |`  C ) )  =  ( ( `' B  |`  C )  o.  `' A )
8 cnvco 5056 . . . 4  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
94, 7, 83eqtr4g 2493 . . 3  |-  ( dom 
A  C_  C  ->  `' ( A  o.  `' ( `' B  |`  C ) )  =  `' ( A  o.  B ) )
109cnveqd 5048 . 2  |-  ( dom 
A  C_  C  ->  `' `' ( A  o.  `' ( `' B  |`  C ) )  =  `' `' ( A  o.  B ) )
11 relco 5368 . . 3  |-  Rel  ( A  o.  `' ( `' B  |`  C ) )
12 dfrel2 5321 . . 3  |-  ( Rel  ( A  o.  `' ( `' B  |`  C ) )  <->  `' `' ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  `' ( `' B  |`  C ) ) )
1311, 12mpbi 200 . 2  |-  `' `' ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  `' ( `' B  |`  C )
)
14 relco 5368 . . 3  |-  Rel  ( A  o.  B )
15 dfrel2 5321 . . 3  |-  ( Rel  ( A  o.  B
)  <->  `' `' ( A  o.  B )  =  ( A  o.  B ) )
1614, 15mpbi 200 . 2  |-  `' `' ( A  o.  B
)  =  ( A  o.  B )
1710, 13, 163eqtr3g 2491 1  |-  ( dom 
A  C_  C  ->  ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    C_ wss 3320   `'ccnv 4877   dom cdm 4878   ran crn 4879    |` cres 4880    o. ccom 4882   Rel wrel 4883
This theorem is referenced by:  fcoi1  5617
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890
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