Theorem funcnvres 4992

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 4376	. . . 4 |- ( F " A ) = ran ( F |` A )
2		df-rn 4374	. . . 4 |- ran ( F |` A ) = dom `' ( F |` A )
3	1, 2	eqtri 2101	. . 3 |- ( F " A ) = dom `' ( F |` A )
4	3	reseq2i 4627	. 2 |- ( `' F |` ( F " A ) ) = ( `' F |` dom `' ( F |` A ) )
5		resss 4653	. . . 4 |- ( F |` A ) C_ F
6		cnvss 4526	. . . 4 |- ( ( F |` A ) C_ F -> `' ( F |` A ) C_ `' F )
7	5, 6	ax-mp 7	. . 3 |- `' ( F |` A ) C_ `' F
8		funssres 4962	. . 3 |- ( ( Fun `' F /\ `' ( F |` A ) C_ `' F ) -> ( `' F |` dom `' ( F |` A ) ) = `' ( F |` A ) )
9	7, 8	mpan2 415	. 2 |- ( Fun `' F -> ( `' F |` dom `' ( F |` A ) ) = `' ( F |` A ) )
10	4, 9	syl5req 2126	1 |- ( Fun `' F -> `' ( F |` A ) = ( `' F |` ( F " A ) ) )

Syntax hints:   -> wi 4   = wceq 1284   C_ wss 2973   `'ccnv 4362   dom cdm 4363   ran crn 4364   |` cres 4365   "cima 4366   Fun wfun 4916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-fun 4924

This theorem is referenced by:  cnvresid  4993  funcnvres2  4994  f1orescnv  5162  f1imacnv  5163