Theorem relbrcnvg 5504

Description: When R is a relation, the sethood assumptions on brcnv 5305 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
relbrcnvg	|- ( Rel R -> ( A `' R B <-> B R A ) )

Proof of Theorem relbrcnvg

Step	Hyp	Ref	Expression
1		relcnv 5503	. . . 4 |- Rel `' R
2		brrelex12 5155	. . . 4 |- ( ( Rel `' R /\ A `' R B ) -> ( A e. _V /\ B e. _V ) )
3	1, 2	mpan 706	. . 3 |- ( A `' R B -> ( A e. _V /\ B e. _V ) )
4	3	a1i 11	. 2 |- ( Rel R -> ( A `' R B -> ( A e. _V /\ B e. _V ) ) )
5		brrelex12 5155	. . . 4 |- ( ( Rel R /\ B R A ) -> ( B e. _V /\ A e. _V ) )
6	5	ancomd 467	. . 3 |- ( ( Rel R /\ B R A ) -> ( A e. _V /\ B e. _V ) )
7	6	ex 450	. 2 |- ( Rel R -> ( B R A -> ( A e. _V /\ B e. _V ) ) )
8		brcnvg 5303	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( A `' R B <-> B R A ) )
9	8	a1i 11	. 2 |- ( Rel R -> ( ( A e. _V /\ B e. _V ) -> ( A `' R B <-> B R A ) ) )
10	4, 7, 9	pm5.21ndd 369	1 |- ( Rel R -> ( A `' R B <-> B R A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   `'ccnv 5113   Rel wrel 5119

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-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-xp 5120  df-rel 5121  df-cnv 5122

This theorem is referenced by:  eliniseg2  5505  relbrcnv  5506  isinv  16420  releleccnv  34021  relcnveq2  34094  brco2f1o  38330  brco3f1o  38331  ntrclsnvobr  38350  neicvgel1  38417