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Theorem cores2 5526
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 5205 . . . . . 6  |-  dom  A  =  ran  `' A
21sseq1i 3523 . . . . 5  |-  ( dom 
A  C_  C  <->  ran  `' A  C_  C )
3 cores 5516 . . . . 5  |-  ( ran  `' A  C_  C  -> 
( ( `' B  |`  C )  o.  `' A )  =  ( `' B  o.  `' A ) )
42, 3sylbi 195 . . . 4  |-  ( dom 
A  C_  C  ->  ( ( `' B  |`  C )  o.  `' A )  =  ( `' B  o.  `' A ) )
5 cnvco 5198 . . . . 5  |-  `' ( A  o.  `' ( `' B  |`  C ) )  =  ( `' `' ( `' B  |`  C )  o.  `' A )
6 cocnvcnv1 5524 . . . . 5  |-  ( `' `' ( `' B  |`  C )  o.  `' A )  =  ( ( `' B  |`  C )  o.  `' A )
75, 6eqtri 2486 . . . 4  |-  `' ( A  o.  `' ( `' B  |`  C ) )  =  ( ( `' B  |`  C )  o.  `' A )
8 cnvco 5198 . . . 4  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
94, 7, 83eqtr4g 2523 . . 3  |-  ( dom 
A  C_  C  ->  `' ( A  o.  `' ( `' B  |`  C ) )  =  `' ( A  o.  B ) )
109cnveqd 5188 . 2  |-  ( dom 
A  C_  C  ->  `' `' ( A  o.  `' ( `' B  |`  C ) )  =  `' `' ( A  o.  B ) )
11 relco 5511 . . 3  |-  Rel  ( A  o.  `' ( `' B  |`  C ) )
12 dfrel2 5463 . . 3  |-  ( Rel  ( A  o.  `' ( `' B  |`  C ) )  <->  `' `' ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  `' ( `' B  |`  C ) ) )
1311, 12mpbi 208 . 2  |-  `' `' ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  `' ( `' B  |`  C )
)
14 relco 5511 . . 3  |-  Rel  ( A  o.  B )
15 dfrel2 5463 . . 3  |-  ( Rel  ( A  o.  B
)  <->  `' `' ( A  o.  B )  =  ( A  o.  B ) )
1614, 15mpbi 208 . 2  |-  `' `' ( A  o.  B
)  =  ( A  o.  B )
1710, 13, 163eqtr3g 2521 1  |-  ( dom 
A  C_  C  ->  ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    C_ wss 3471   `'ccnv 5007   dom cdm 5008   ran crn 5009    |` cres 5010    o. ccom 5012   Rel wrel 5013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020
This theorem is referenced by:  fcoi1  5765  ofco2  19079
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