Theorem funcnvres2 5456

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 5440	. . 3 |- ( Fun F -> Fun `' `' F )
2		funcnvres 5454	. . 3 |- ( Fun `' `' F -> `' ( `' F |` A ) = ( `' `' F |` ( `' F " A ) ) )
3	1, 2	syl 14	. 2 |- ( Fun F -> `' ( `' F |` A ) = ( `' `' F |` ( `' F " A ) ) )
4		funrel 5394	. . . 4 |- ( Fun F -> Rel F )
5		dfrel2 5238	. . . 4 |- ( Rel F <-> `' `' F = F )
6	4, 5	sylib 122	. . 3 |- ( Fun F -> `' `' F = F )
7	6	reseq1d 5062	. 2 |- ( Fun F -> ( `' `' F |` ( `' F " A ) ) = ( F |` ( `' F " A ) ) )
8	3, 7	eqtrd 2271	1 |- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   `'ccnv 4773   |` cres 4776   "cima 4777   Rel wrel 4779   Fun wfun 5371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-fun 5379

This theorem is used by:  funimacnv  5457  foimacnv  5657