Theorem funcnvres 4972

Description: The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)

Assertion

Ref	Expression
funcnvres	|- ( Fun `' F -> `' ( F |` A ) = ( `' F |` ( F " A ) ) )

Proof of Theorem funcnvres

Step	Hyp	Ref	Expression
1		df-ima 4358	. . . 4 |- ( F " A ) = ran ( F |` A )
2		df-rn 4356	. . . 4 |- ran ( F |` A ) = dom `' ( F |` A )
3	1, 2	eqtri 2060	. . 3 |- ( F " A ) = dom `' ( F |` A )
4	3	reseq2i 4609	. 2 |- ( `' F |` ( F " A ) ) = ( `' F |` dom `' ( F |` A ) )
5		resss 4635	. . . 4 |- ( F |` A ) C_ F
6		cnvss 4508	. . . 4 |- ( ( F |` A ) C_ F -> `' ( F |` A ) C_ `' F )
7	5, 6	ax-mp 7	. . 3 |- `' ( F |` A ) C_ `' F
8		funssres 4942	. . 3 |- ( ( Fun `' F /\ `' ( F |` A ) C_ `' F ) -> ( `' F |` dom `' ( F |` A ) ) = `' ( F |` A ) )
9	7, 8	mpan2 401	. 2 |- ( Fun `' F -> ( `' F |` dom `' ( F |` A ) ) = `' ( F |` A ) )
10	4, 9	syl5req 2085	1 |- ( Fun `' F -> `' ( F |` A ) = ( `' F |` ( F " A ) ) )

Syntax hints:   -> wi 4   = wceq 1243   C_ wss 2917   `'ccnv 4344   dom cdm 4345   ran crn 4346   |` cres 4347   "cima 4348   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:  cnvresid  4973  funcnvres2  4974  f1orescnv  5142  f1imacnv  5143