Theorem relbrcnvg 4913

Description: When R is a relation, the sethood assumptions on brcnv 4717 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 4912	. . . 4 |- Rel `' R
2		brrelex12 4572	. . . 4 |- ( ( Rel `' R /\ A `' R B ) -> ( A e. _V /\ B e. _V ) )
3	1, 2	mpan 420	. . 3 |- ( A `' R B -> ( A e. _V /\ B e. _V ) )
4	3	a1i 9	. 2 |- ( Rel R -> ( A `' R B -> ( A e. _V /\ B e. _V ) ) )
5		brrelex12 4572	. . . 4 |- ( ( Rel R /\ B R A ) -> ( B e. _V /\ A e. _V ) )
6	5	ancomd 265	. . 3 |- ( ( Rel R /\ B R A ) -> ( A e. _V /\ B e. _V ) )
7	6	ex 114	. 2 |- ( Rel R -> ( B R A -> ( A e. _V /\ B e. _V ) ) )
8		brcnvg 4715	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( A `' R B <-> B R A ) )
9	8	a1i 9	. 2 |- ( Rel R -> ( ( A e. _V /\ B e. _V ) -> ( A `' R B <-> B R A ) ) )
10	4, 7, 9	pm5.21ndd 694	1 |- ( Rel R -> ( A `' R B <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1480   _Vcvv 2681   class class class wbr 3924   `'ccnv 4533   Rel wrel 4539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542

This theorem is referenced by:  relbrcnv  4914