Theorem funcnvres 5291

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 4641	. . . 4 |- ( F " A ) = ran ( F |` A )
2		df-rn 4639	. . . 4 |- ran ( F |` A ) = dom `' ( F |` A )
3	1, 2	eqtri 2198	. . 3 |- ( F " A ) = dom `' ( F |` A )
4	3	reseq2i 4906	. 2 |- ( `' F |` ( F " A ) ) = ( `' F |` dom `' ( F |` A ) )
5		resss 4933	. . . 4 |- ( F |` A ) C_ F
6		cnvss 4802	. . . 4 |- ( ( F |` A ) C_ F -> `' ( F |` A ) C_ `' F )
7	5, 6	ax-mp 5	. . 3 |- `' ( F |` A ) C_ `' F
8		funssres 5260	. . 3 |- ( ( Fun `' F /\ `' ( F |` A ) C_ `' F ) -> ( `' F |` dom `' ( F |` A ) ) = `' ( F |` A ) )
9	7, 8	mpan2 425	. 2 |- ( Fun `' F -> ( `' F |` dom `' ( F |` A ) ) = `' ( F |` A ) )
10	4, 9	eqtr2id 2223	1 |- ( Fun `' F -> `' ( F |` A ) = ( `' F |` ( F " A ) ) )

Syntax hints:   -> wi 4   = wceq 1353   C_ wss 3131   `'ccnv 4627   dom cdm 4628   ran crn 4629   |` cres 4630   "cima 4631   Fun wfun 5212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-fun 5220

This theorem is referenced by:  cnvresid  5292  funcnvres2  5293  f1orescnv  5479  f1imacnv  5480  sbthlemi4  6961  hmeores  13900