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Theorem cores2 5201
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 4888 . . . . . 6  |-  dom  A  =  ran  `' A
21sseq1i 3215 . . . . 5  |-  ( dom 
A  C_  C  <->  ran  `' A  C_  C )
3 cores 5192 . . . . 5  |-  ( ran  `' A  C_  C  -> 
( ( `' B  |`  C )  o.  `' A )  =  ( `' B  o.  `' A ) )
42, 3sylbi 187 . . . 4  |-  ( dom 
A  C_  C  ->  ( ( `' B  |`  C )  o.  `' A )  =  ( `' B  o.  `' A ) )
5 cnvco 4881 . . . . 5  |-  `' ( A  o.  `' ( `' B  |`  C ) )  =  ( `' `' ( `' B  |`  C )  o.  `' A )
6 cocnvcnv1 5199 . . . . 5  |-  ( `' `' ( `' B  |`  C )  o.  `' A )  =  ( ( `' B  |`  C )  o.  `' A )
75, 6eqtri 2316 . . . 4  |-  `' ( A  o.  `' ( `' B  |`  C ) )  =  ( ( `' B  |`  C )  o.  `' A )
8 cnvco 4881 . . . 4  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
94, 7, 83eqtr4g 2353 . . 3  |-  ( dom 
A  C_  C  ->  `' ( A  o.  `' ( `' B  |`  C ) )  =  `' ( A  o.  B ) )
109cnveqd 4873 . 2  |-  ( dom 
A  C_  C  ->  `' `' ( A  o.  `' ( `' B  |`  C ) )  =  `' `' ( A  o.  B ) )
11 relco 5187 . . 3  |-  Rel  ( A  o.  `' ( `' B  |`  C ) )
12 dfrel2 5140 . . 3  |-  ( Rel  ( A  o.  `' ( `' B  |`  C ) )  <->  `' `' ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  `' ( `' B  |`  C ) ) )
1311, 12mpbi 199 . 2  |-  `' `' ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  `' ( `' B  |`  C )
)
14 relco 5187 . . 3  |-  Rel  ( A  o.  B )
15 dfrel2 5140 . . 3  |-  ( Rel  ( A  o.  B
)  <->  `' `' ( A  o.  B )  =  ( A  o.  B ) )
1614, 15mpbi 199 . 2  |-  `' `' ( A  o.  B
)  =  ( A  o.  B )
1710, 13, 163eqtr3g 2351 1  |-  ( dom 
A  C_  C  ->  ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    C_ wss 3165   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707    o. ccom 4709   Rel wrel 4710
This theorem is referenced by:  fcoi1  5431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717
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