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Theorem cores2 4930
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 4616 . . . . . 6  |-  dom  A  =  ran  `' A
21sseq1i 3048 . . . . 5  |-  ( dom 
A  C_  C  <->  ran  `' A  C_  C )
3 cores 4921 . . . . 5  |-  ( ran  `' A  C_  C  -> 
( ( `' B  |`  C )  o.  `' A )  =  ( `' B  o.  `' A ) )
42, 3sylbi 119 . . . 4  |-  ( dom 
A  C_  C  ->  ( ( `' B  |`  C )  o.  `' A )  =  ( `' B  o.  `' A ) )
5 cnvco 4609 . . . . 5  |-  `' ( A  o.  `' ( `' B  |`  C ) )  =  ( `' `' ( `' B  |`  C )  o.  `' A )
6 cocnvcnv1 4928 . . . . 5  |-  ( `' `' ( `' B  |`  C )  o.  `' A )  =  ( ( `' B  |`  C )  o.  `' A )
75, 6eqtri 2108 . . . 4  |-  `' ( A  o.  `' ( `' B  |`  C ) )  =  ( ( `' B  |`  C )  o.  `' A )
8 cnvco 4609 . . . 4  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
94, 7, 83eqtr4g 2145 . . 3  |-  ( dom 
A  C_  C  ->  `' ( A  o.  `' ( `' B  |`  C ) )  =  `' ( A  o.  B ) )
109cnveqd 4600 . 2  |-  ( dom 
A  C_  C  ->  `' `' ( A  o.  `' ( `' B  |`  C ) )  =  `' `' ( A  o.  B ) )
11 relco 4916 . . 3  |-  Rel  ( A  o.  `' ( `' B  |`  C ) )
12 dfrel2 4868 . . 3  |-  ( Rel  ( A  o.  `' ( `' B  |`  C ) )  <->  `' `' ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  `' ( `' B  |`  C ) ) )
1311, 12mpbi 143 . 2  |-  `' `' ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  `' ( `' B  |`  C )
)
14 relco 4916 . . 3  |-  Rel  ( A  o.  B )
15 dfrel2 4868 . . 3  |-  ( Rel  ( A  o.  B
)  <->  `' `' ( A  o.  B )  =  ( A  o.  B ) )
1614, 15mpbi 143 . 2  |-  `' `' ( A  o.  B
)  =  ( A  o.  B )
1710, 13, 163eqtr3g 2143 1  |-  ( dom 
A  C_  C  ->  ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    C_ wss 2997   `'ccnv 4427   dom cdm 4428   ran crn 4429    |` cres 4430    o. ccom 4432   Rel wrel 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440
This theorem is referenced by:  cocnvres  4942  fcoi1  5175
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