Theorem funcnvres 5154

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 4512	. . . 4 |- ( F " A ) = ran ( F |` A )
2		df-rn 4510	. . . 4 |- ran ( F |` A ) = dom `' ( F |` A )
3	1, 2	eqtri 2135	. . 3 |- ( F " A ) = dom `' ( F |` A )
4	3	reseq2i 4774	. 2 |- ( `' F |` ( F " A ) ) = ( `' F |` dom `' ( F |` A ) )
5		resss 4801	. . . 4 |- ( F |` A ) C_ F
6		cnvss 4672	. . . 4 |- ( ( F |` A ) C_ F -> `' ( F |` A ) C_ `' F )
7	5, 6	ax-mp 7	. . 3 |- `' ( F |` A ) C_ `' F
8		funssres 5123	. . 3 |- ( ( Fun `' F /\ `' ( F |` A ) C_ `' F ) -> ( `' F |` dom `' ( F |` A ) ) = `' ( F |` A ) )
9	7, 8	mpan2 419	. 2 |- ( Fun `' F -> ( `' F |` dom `' ( F |` A ) ) = `' ( F |` A ) )
10	4, 9	syl5req 2160	1 |- ( Fun `' F -> `' ( F |` A ) = ( `' F |` ( F " A ) ) )

Syntax hints:   -> wi 4   = wceq 1314   C_ wss 3037   `'ccnv 4498   dom cdm 4499   ran crn 4500   |` cres 4501   "cima 4502   Fun wfun 5075

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-fun 5083

This theorem is referenced by:  cnvresid  5155  funcnvres2  5156  f1orescnv  5339  f1imacnv  5340  sbthlemi4  6800