Theorem funcnvres 3574

Description: The converse of a restricted function.

Assertion

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

Proof of Theorem funcnvres

Step	Hyp	Ref	Expression
1		resss 3389	. . . 4 |- (F |` A) (_ F
2		cnvss 3297	. . . 4 |- ((F |` A) (_ F -> `'(F |` A) (_ `'F)
3	1, 2	ax-mp 7	. . 3 |- `'(F |` A) (_ `'F
4		funssres 3558	. . 3 |- ((Fun `'F /\ `'(F |` A) (_ `'F) -> (`'F |` dom `'(F |` A)) = `'(F |` A))
5	3, 4	mpan2 698	. 2 |- (Fun `'F -> (`'F |` dom `'(F |` A)) = `'(F |` A))
6		df-ima 3197	. . . 4 |- (F"A) = ran ( F |` A)
7		df-rn 3195	. . . 4 |- ran ( F |` A) = dom `'(F |` A)
8	6, 7	eqtr 1498	. . 3 |- (F"A) = dom `'(F |` A)
9		reseq2 3375	. . 3 |- ((F"A) = dom `'(F |` A) -> (`'F |` (F"A)) = (`'F |` dom `'(F |` A)))
10	8, 9	ax-mp 7	. 2 |- (`'F |` (F"A)) = (`'F |` dom `'(F |` A))
11	5, 10	syl5req 1523	1 |- (Fun `'F -> `'(F |` A) = (`'F |` (F"A)))

Syntax hints:   -> wi 3   = wceq 958   (_ wss 2050  `'ccnv 3175  dom cdm 3176  ran crn 3177   |` cres 3178  "cima 3179  Fun wfun 3182

This theorem is referenced by:  cnvresid 3575  funcnvres2 3576  f1orescnv 3710  f1imacnv 3711  sbthlem4 4456  idcn 7763  dfrelog 8751

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198

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