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Theorem funcnvres2 5005
Description: The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.)
Assertion
Ref Expression
funcnvres2  |-  ( Fun 
F  ->  `' ( `' F  |`  A )  =  ( F  |`  ( `' F " A ) ) )

Proof of Theorem funcnvres2
StepHypRef Expression
1 funcnvcnv 4989 . . 3  |-  ( Fun 
F  ->  Fun  `' `' F )
2 funcnvres 5003 . . 3  |-  ( Fun  `' `' F  ->  `' ( `' F  |`  A )  =  ( `' `' F  |`  ( `' F " A ) ) )
31, 2syl 14 . 2  |-  ( Fun 
F  ->  `' ( `' F  |`  A )  =  ( `' `' F  |`  ( `' F " A ) ) )
4 funrel 4949 . . . 4  |-  ( Fun 
F  ->  Rel  F )
5 dfrel2 4801 . . . 4  |-  ( Rel 
F  <->  `' `' F  =  F
)
64, 5sylib 120 . . 3  |-  ( Fun 
F  ->  `' `' F  =  F )
76reseq1d 4639 . 2  |-  ( Fun 
F  ->  ( `' `' F  |`  ( `' F " A ) )  =  ( F  |`  ( `' F " A ) ) )
83, 7eqtrd 2114 1  |-  ( Fun 
F  ->  `' ( `' F  |`  A )  =  ( F  |`  ( `' F " A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   `'ccnv 4370    |` cres 4373   "cima 4374   Rel wrel 4376   Fun wfun 4926
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-fun 4934
This theorem is referenced by:  funimacnv  5006  foimacnv  5175
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