Theorem relbrcnvg 4983

Description: When R is a relation, the sethood assumptions on brcnv 4787 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 4982	. . . 4 |- Rel `' R
2		brrelex12 4642	. . . 4 |- ( ( Rel `' R /\ A `' R B ) -> ( A e. _V /\ B e. _V ) )
3	1, 2	mpan 421	. . 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 4642	. . . 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 4785	. . 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 695	1 |- ( Rel R -> ( A `' R B <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2136   _Vcvv 2726   class class class wbr 3982   `'ccnv 4603   Rel wrel 4609

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612

This theorem is referenced by:  relbrcnv  4984