Theorem brcnvepres 34033

Description: Restricted converse epsilon binary relation. (Contributed by Peter Mazsa, 10-Feb-2018.)

Assertion

Ref	Expression
brcnvepres	|- ( ( B e. V /\ C e. W ) -> ( B ( `' _E |` A ) C <-> ( B e. A /\ C e. B ) ) )

Proof of Theorem brcnvepres

Step	Hyp	Ref	Expression
1		brresALTV 34032	. 2 |- ( C e. W -> ( B ( `' _E |` A ) C <-> ( B e. A /\ B `' _E C ) ) )
2		brcnvep 34029	. . 3 |- ( B e. V -> ( B `' _E C <-> C e. B ) )
3	2	anbi2d 740	. 2 |- ( B e. V -> ( ( B e. A /\ B `' _E C ) <-> ( B e. A /\ C e. B ) ) )
4	1, 3	sylan9bbr 737	1 |- ( ( B e. V /\ C e. W ) -> ( B ( `' _E |` A ) C <-> ( B e. A /\ C e. B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   class class class wbr 4653   _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: (None)