Theorem cnvepres 34066

Description: Restricted converse epsilon relation as a class of ordered pairs. (Contributed by Peter Mazsa, 10-Feb-2018.)

Assertion

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

Distinct variable group:   x, A, y

Proof of Theorem cnvepres

Step	Hyp	Ref	Expression
1		dfres2 5453	. 2 |- ( `' _E |` A ) = { <. x , y >. | ( x e. A /\ x `' _E y ) }
2		brcnvep 34029	. . . . 5 |- ( x e. _V -> ( x `' _E y <-> y e. x ) )
3	2	elv 33983	. . . 4 |- ( x `' _E y <-> y e. x )
4	3	anbi2i 730	. . 3 |- ( ( x e. A /\ x `' _E y ) <-> ( x e. A /\ y e. x ) )
5	4	opabbii 4717	. 2 |- { <. x , y >. | ( x e. A /\ x `' _E y ) } = { <. x , y >. | ( x e. A /\ y e. x ) }
6	1, 5	eqtri 2644	1 |- ( `' _E |` A ) = { <. x , y >. | ( x e. A /\ y e. x ) }

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   {copab 4712   _E cep 5028   `'ccnv 5113   |` 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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  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-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-res 5126

This theorem is referenced by:  rncnvepres  34073  n0el2  34103  cnvepresex  34104