Theorem funcnvres2 4974

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 4958	. . 3 |- ( Fun F -> Fun `' `' F )
2		funcnvres 4972	. . 3 |- ( Fun `' `' F -> `' ( `' F |` A ) = ( `' `' F |` ( `' F " A ) ) )
3	1, 2	syl 14	. 2 |- ( Fun F -> `' ( `' F |` A ) = ( `' `' F |` ( `' F " A ) ) )
4		funrel 4919	. . . 4 |- ( Fun F -> Rel F )
5		dfrel2 4771	. . . 4 |- ( Rel F <-> `' `' F = F )
6	4, 5	sylib 127	. . 3 |- ( Fun F -> `' `' F = F )
7	6	reseq1d 4611	. 2 |- ( Fun F -> ( `' `' F |` ( `' F " A ) ) = ( F |` ( `' F " A ) ) )
8	3, 7	eqtrd 2072	1 |- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) )

Syntax hints:   -> wi 4   = wceq 1243   `'ccnv 4344   |` cres 4347   "cima 4348   Rel wrel 4350   Fun wfun 4896

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fun 4904

This theorem is referenced by:  funimacnv  4975  foimacnv  5144