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Theorem funcnvres2 5089
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 5073 . . 3  |-  ( Fun 
F  ->  Fun  `' `' F )
2 funcnvres 5087 . . 3  |-  ( Fun  `' `' F  ->  `' ( `' F  |`  A )  =  ( `' `' F  |`  ( `' F " A ) ) )
31, 2syl 14 . 2  |-  ( Fun 
F  ->  `' ( `' F  |`  A )  =  ( `' `' F  |`  ( `' F " A ) ) )
4 funrel 5032 . . . 4  |-  ( Fun 
F  ->  Rel  F )
5 dfrel2 4881 . . . 4  |-  ( Rel 
F  <->  `' `' F  =  F
)
64, 5sylib 120 . . 3  |-  ( Fun 
F  ->  `' `' F  =  F )
76reseq1d 4712 . 2  |-  ( Fun 
F  ->  ( `' `' F  |`  ( `' F " A ) )  =  ( F  |`  ( `' F " A ) ) )
83, 7eqtrd 2120 1  |-  ( Fun 
F  ->  `' ( `' F  |`  A )  =  ( F  |`  ( `' F " A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   `'ccnv 4437    |` cres 4440   "cima 4441   Rel wrel 4443   Fun wfun 5009
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 3957  ax-pow 4009  ax-pr 4036
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 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-fun 5017
This theorem is referenced by:  funimacnv  5090  foimacnv  5271
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