Theorem opabresid 4960

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 4951	. 2 |- ( { <. x , y >. | y = x } |` A ) = { <. x , y >. | ( x e. A /\ y = x ) }
2		equcom 1706	. . . . 5 |- ( y = x <-> x = y )
3	2	opabbii 4070	. . . 4 |- { <. x , y >. | y = x } = { <. x , y >. | x = y }
4		df-id 4293	. . . 4 |- _I = { <. x , y >. | x = y }
5	3, 4	eqtr4i 2201	. . 3 |- { <. x , y >. | y = x } = _I
6	5	reseq1i 4903	. 2 |- ( { <. x , y >. | y = x } |` A ) = ( _I |` A )
7	1, 6	eqtr3i 2200	1 |- { <. x , y >. | ( x e. A /\ y = x ) } = ( _I |` A )

Syntax hints:   /\ wa 104   = wceq 1353   e. wcel 2148   {copab 4063   _I cid 4288   |` cres 4628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-res 4638

This theorem is referenced by:  mptresid  4961