Theorem opabresid 5455

Description: The restricted identity expressed with the class builder. (Contributed by FL, 25-Apr-2012.)

Assertion

Ref	Expression
opabresid	|- { <. x , y >. | ( x e. A /\ y = x ) } = ( _I |` A )

Distinct variable group:   x, A, y

Proof of Theorem opabresid

Step	Hyp	Ref	Expression
1		resopab 5446	. 2 |- ( { <. x , y >. | y = x } |` A ) = { <. x , y >. | ( x e. A /\ y = x ) }
2		equcom 1945	. . . . 5 |- ( y = x <-> x = y )
3	2	opabbii 4717	. . . 4 |- { <. x , y >. | y = x } = { <. x , y >. | x = y }
4		dfid3 5025	. . . 4 |- _I = { <. x , y >. | x = y }
5	3, 4	eqtr4i 2647	. . 3 |- { <. x , y >. | y = x } = _I
6	5	reseq1i 5392	. 2 |- ( { <. x , y >. | y = x } |` A ) = ( _I |` A )
7	1, 6	eqtr3i 2646	1 |- { <. x , y >. | ( x e. A /\ y = x ) } = ( _I |` A )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {copab 4712   _I cid 5023   |` cres 5116

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-res 5126

This theorem is referenced by:  mptresid  5456  pospo  16973  opsrtoslem1  19484  tgphaus  21920  relexp0eq  37993