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Theorem funcnvres2 5526
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 5511 . . 3  |-  ( Fun 
F  ->  Fun  `' `' F )
2 funcnvres 5524 . . 3  |-  ( Fun  `' `' F  ->  `' ( `' F  |`  A )  =  ( `' `' F  |`  ( `' F " A ) ) )
31, 2syl 16 . 2  |-  ( Fun 
F  ->  `' ( `' F  |`  A )  =  ( `' `' F  |`  ( `' F " A ) ) )
4 funrel 5473 . . . 4  |-  ( Fun 
F  ->  Rel  F )
5 dfrel2 5323 . . . 4  |-  ( Rel 
F  <->  `' `' F  =  F
)
64, 5sylib 190 . . 3  |-  ( Fun 
F  ->  `' `' F  =  F )
76reseq1d 5147 . 2  |-  ( Fun 
F  ->  ( `' `' F  |`  ( `' F " A ) )  =  ( F  |`  ( `' F " A ) ) )
83, 7eqtrd 2470 1  |-  ( Fun 
F  ->  `' ( `' F  |`  A )  =  ( F  |`  ( `' F " A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   `'ccnv 4879    |` cres 4882   "cima 4883   Rel wrel 4885   Fun wfun 5450
This theorem is referenced by:  funimacnv  5527  foimacnv  5694  unbenlem  13278  dvlog  20544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-fun 5458
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