Theorem funcnvres 5356

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 4696	. . . 4 |- ( F " A ) = ran ( F |` A )
2		df-rn 4694	. . . 4 |- ran ( F |` A ) = dom `' ( F |` A )
3	1, 2	eqtri 2227	. . 3 |- ( F " A ) = dom `' ( F |` A )
4	3	reseq2i 4965	. 2 |- ( `' F |` ( F " A ) ) = ( `' F |` dom `' ( F |` A ) )
5		resss 4992	. . . 4 |- ( F |` A ) C_ F
6		cnvss 4859	. . . 4 |- ( ( F |` A ) C_ F -> `' ( F |` A ) C_ `' F )
7	5, 6	ax-mp 5	. . 3 |- `' ( F |` A ) C_ `' F
8		funssres 5322	. . 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 2252	1 |- ( Fun `' F -> `' ( F |` A ) = ( `' F |` ( F " A ) ) )

Syntax hints:   -> wi 4   = wceq 1373   C_ wss 3170   `'ccnv 4682   dom cdm 4683   ran crn 4684   |` cres 4685   "cima 4686   Fun wfun 5274

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-fun 5282

This theorem is referenced by:  cnvresid  5357  funcnvres2  5358  f1orescnv  5550  f1imacnv  5551  sbthlemi4  7077  hmeores  14862