Theorem cnvresid 3575

Description: Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)

Assertion

Ref	Expression
cnvresid	|- `'(I |` A) = (I |` A)

Proof of Theorem cnvresid

Step	Hyp	Ref	Expression
1		cnvi 3453	. . . 4 |- `'I = I
2	1	eqcomi 1482	. . 3 |- I = `'I
3		funi 3551	. . . 4 |- Fun I
4		funeq 3541	. . . 4 |- (I = `'I -> (Fun I <-> Fun `'I))
5	3, 4	mpbii 193	. . 3 |- (I = `'I -> Fun `'I)
6	2, 5	ax-mp 7	. 2 |- Fun `'I
7		funcnvres 3574	. . 3 |- (Fun `'I -> `'(I |` A) = (`'I |` (I"A)))
8		reseq1 3374	. . . . 5 |- (`'I = I -> (`'I |` (I"A)) = (I |` (I"A)))
9		imai 3423	. . . . . 6 |- (I"A) = A
10		reseq2 3375	. . . . . 6 |- ((I"A) = A -> (I |` (I"A)) = (I |` A))
11	9, 10	ax-mp 7	. . . . 5 |- (I |` (I"A)) = (I |` A)
12	8, 11	syl6eq 1526	. . . 4 |- (`'I = I -> (`'I |` (I"A)) = (I |` A))
13	1, 12	ax-mp 7	. . 3 |- (`'I |` (I"A)) = (I |` A)
14	7, 13	syl6eq 1526	. 2 |- (Fun `'I -> `'(I |` A) = (I |` A))
15	6, 14	ax-mp 7	1 |- `'(I |` A) = (I |` A)

Syntax hints:   = wceq 958  Icid 2837  `'ccnv 3175   |` cres 3178  "cima 3179  Fun wfun 3182

This theorem is referenced by:  idhme 10508  hmphre 10516

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-rex 1653  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

Copyright terms: Public domain