Theorem relbrcnvg 5062

Description: When R is a relation, the sethood assumptions on brcnv 4862 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 5061	. . . 4 |- Rel `' R
2		brrelex12 4714	. . . 4 |- ( ( Rel `' R /\ A `' R B ) -> ( A e. _V /\ B e. _V ) )
3	1, 2	mpan 424	. . 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 4714	. . . 4 |- ( ( Rel R /\ B R A ) -> ( B e. _V /\ A e. _V ) )
6	5	ancomd 267	. . 3 |- ( ( Rel R /\ B R A ) -> ( A e. _V /\ B e. _V ) )
7	6	ex 115	. 2 |- ( Rel R -> ( B R A -> ( A e. _V /\ B e. _V ) ) )
8		brcnvg 4860	. . 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 707	1 |- ( Rel R -> ( A `' R B <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2176   _Vcvv 2772   class class class wbr 4045   `'ccnv 4675   Rel wrel 4681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-xp 4682  df-rel 4683  df-cnv 4684

This theorem is referenced by:  eliniseg2  5063  relbrcnv  5064