Theorem relbrcnvg 4764

Description: When R is a relation, the sethood assumptions on brcnv 4575 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 4763	. . . 4 |- Rel `' R
2		brrelex12 4435	. . . 4 |- ( ( Rel `' R /\ A `' R B ) -> ( A e. _V /\ B e. _V ) )
3	1, 2	mpan 415	. . 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 4435	. . . 4 |- ( ( Rel R /\ B R A ) -> ( B e. _V /\ A e. _V ) )
6	5	ancomd 263	. . 3 |- ( ( Rel R /\ B R A ) -> ( A e. _V /\ B e. _V ) )
7	6	ex 113	. 2 |- ( Rel R -> ( B R A -> ( A e. _V /\ B e. _V ) ) )
8		brcnvg 4573	. . 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 654	1 |- ( Rel R -> ( A `' R B <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1434   _Vcvv 2612   class class class wbr 3811   `'ccnv 4398   Rel wrel 4404

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-xp 4405  df-rel 4406  df-cnv 4407

This theorem is referenced by:  relbrcnv  4765