MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cores2 Unicode version

Theorem cores2 5185
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 4872 . . . . . 6  |-  dom  A  =  ran  `' A
21sseq1i 3202 . . . . 5  |-  ( dom 
A  C_  C  <->  ran  `' A  C_  C )
3 cores 5176 . . . . 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 4865 . . . . 5  |-  `' ( A  o.  `' ( `' B  |`  C ) )  =  ( `' `' ( `' B  |`  C )  o.  `' A )
6 cocnvcnv1 5183 . . . . 5  |-  ( `' `' ( `' B  |`  C )  o.  `' A )  =  ( ( `' B  |`  C )  o.  `' A )
75, 6eqtri 2303 . . . 4  |-  `' ( A  o.  `' ( `' B  |`  C ) )  =  ( ( `' B  |`  C )  o.  `' A )
8 cnvco 4865 . . . 4  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
94, 7, 83eqtr4g 2340 . . 3  |-  ( dom 
A  C_  C  ->  `' ( A  o.  `' ( `' B  |`  C ) )  =  `' ( A  o.  B ) )
109cnveqd 4857 . 2  |-  ( dom 
A  C_  C  ->  `' `' ( A  o.  `' ( `' B  |`  C ) )  =  `' `' ( A  o.  B ) )
11 relco 5171 . . 3  |-  Rel  ( A  o.  `' ( `' B  |`  C ) )
12 dfrel2 5124 . . 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 5171 . . 3  |-  Rel  ( A  o.  B )
15 dfrel2 5124 . . 3  |-  ( Rel  ( A  o.  B
)  <->  `' `' ( A  o.  B )  =  ( A  o.  B ) )
1614, 15mpbi 199 . 2  |-  `' `' ( A  o.  B
)  =  ( A  o.  B )
1710, 13, 163eqtr3g 2338 1  |-  ( dom 
A  C_  C  ->  ( A  o.  `' ( `' B  |`  C ) )  =  ( A  o.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    C_ wss 3152   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691    o. ccom 4693   Rel wrel 4694
This theorem is referenced by:  fcoi1  5415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701
  Copyright terms: Public domain W3C validator