Theorem cnvresid 5968

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 5537	. . 3 |- `' _I = _I
2	1	eqcomi 2631	. 2 |- _I = `' _I
3		funi 5920	. . 3 |- Fun _I
4		funeq 5908	. . 3 |- ( _I = `' _I -> ( Fun _I <-> Fun `' _I ) )
5	3, 4	mpbii 223	. 2 |- ( _I = `' _I -> Fun `' _I )
6		funcnvres 5967	. . 3 |- ( Fun `' _I -> `' ( _I |` A ) = ( `' _I |` ( _I " A ) ) )
7		imai 5478	. . . 4 |- ( _I " A ) = A
8	1, 7	reseq12i 5394	. . 3 |- ( `' _I |` ( _I " A ) ) = ( _I |` A )
9	6, 8	syl6eq 2672	. 2 |- ( Fun `' _I -> `' ( _I |` A ) = ( _I |` A ) )
10	2, 5, 9	mp2b 10	1 |- `' ( _I |` A ) = ( _I |` A )

Syntax hints:   = wceq 1483   _I cid 5023   `'ccnv 5113   |` cres 5116   "cima 5117   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890

This theorem is referenced by:  fcoi1  6078  f1oi  6174  relexpcnv  13775  tsrdir  17238  gicref  17713  ssidcn  21059  idqtop  21509  idhmeo  21576  ltrncnvnid  35413  dihmeetlem1N  36579  dihglblem5apreN  36580  diophrw  37322  cnvrcl0  37932  relexpaddss  38010