Theorem coepr 31642

Description: Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)

Hypotheses

Ref	Expression
coep.1	|- A e. _V
coep.2	|- B e. _V

Assertion

Ref	Expression
coepr	|- ( A ( R o. `' _E ) B <-> E. x e. A x R B )

Distinct variable groups:   x, A   x, B   x, R

Proof of Theorem coepr

Step	Hyp	Ref	Expression
1		coep.1	. . . . . 6 |- A e. _V
2		vex 3203	. . . . . 6 |- x e. _V
3	1, 2	brcnv 5305	. . . . 5 |- ( A `' _E x <-> x _E A )
4	1	epelc 5031	. . . . 5 |- ( x _E A <-> x e. A )
5	3, 4	bitri 264	. . . 4 |- ( A `' _E x <-> x e. A )
6	5	anbi1i 731	. . 3 |- ( ( A `' _E x /\ x R B ) <-> ( x e. A /\ x R B ) )
7	6	exbii 1774	. 2 |- ( E. x ( A `' _E x /\ x R B ) <-> E. x ( x e. A /\ x R B ) )
8		coep.2	. . 3 |- B e. _V
9	1, 8	brco 5292	. 2 |- ( A ( R o. `' _E ) B <-> E. x ( A `' _E x /\ x R B ) )
10		df-rex 2918	. 2 |- ( E. x e. A x R B <-> E. x ( x e. A /\ x R B ) )
11	7, 9, 10	3bitr4i 292	1 |- ( A ( R o. `' _E ) B <-> E. x e. A x R B )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   _E cep 5028   `'ccnv 5113   o. ccom 5118

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-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-cnv 5122  df-co 5123

This theorem is referenced by:  elfuns  32022  brub  32061