Theorem funcnvres 3568

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 3383	. . . 4 |- (F |` A) (_ F
2		cnvss 3291	. . . 4 |- ((F |` A) (_ F -> `'(F |` A) (_ `'F)
3	1, 2	ax-mp 7	. . 3 |- `'(F |` A) (_ `'F
4		funssres 3552	. . 3 |- ((Fun `'F /\ `'(F |` A) (_ `'F) -> (`'F |` dom `'(F |` A)) = `'(F |` A))
5	3, 4	mpan2 696	. 2 |- (Fun `'F -> (`'F |` dom `'(F |` A)) = `'(F |` A))
6		df-ima 3191	. . . 4 |- (F"A) = ran ( F |` A)
7		df-rn 3189	. . . 4 |- ran ( F |` A) = dom `'(F |` A)
8	6, 7	eqtr 1495	. . 3 |- (F"A) = dom `'(F |` A)
9		reseq2 3369	. . 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 1520	1 |- (Fun `'F -> `'(F |` A) = (`'F |` (F"A)))

Syntax hints:   -> wi 3   = wceq 956   (_ wss 2047  `'ccnv 3169  dom cdm 3170  ran crn 3171   |` cres 3172  "cima 3173  Fun wfun 3176

This theorem is referenced by:  cnvresid 3569  funcnvres2 3570  f1orescnv 3704  f1imacnv 3705  sbthlem4 4450  idcn 7766  dfrelog 8756

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192

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