Theorem funcnvres2 5436

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

Step	Hyp	Ref	Expression
1		funcnvcnv 5420	. . 3 |- ( Fun F -> Fun `' `' F )
2		funcnvres 5434	. . 3 |- ( Fun `' `' F -> `' ( `' F |` A ) = ( `' `' F |` ( `' F " A ) ) )
3	1, 2	syl 14	. 2 |- ( Fun F -> `' ( `' F |` A ) = ( `' `' F |` ( `' F " A ) ) )
4		funrel 5374	. . . 4 |- ( Fun F -> Rel F )
5		dfrel2 5218	. . . 4 |- ( Rel F <-> `' `' F = F )
6	4, 5	sylib 122	. . 3 |- ( Fun F -> `' `' F = F )
7	6	reseq1d 5042	. 2 |- ( Fun F -> ( `' `' F |` ( `' F " A ) ) = ( F |` ( `' F " A ) ) )
8	3, 7	eqtrd 2267	1 |- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) )

Syntax hints:   -> wi 4   = wceq 1398   `'ccnv 4753   |` cres 4756   "cima 4757   Rel wrel 4759   Fun wfun 5351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-fun 5359

This theorem is referenced by:  funimacnv  5437  foimacnv  5637