Theorem opelcnvg 4515

Description: Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
opelcnvg	|- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. `' R <-> <. B , A >. e. R ) )

Proof of Theorem opelcnvg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 3768	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 3767	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 4353	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4006	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 3765	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 3765	. 2 |- ( B R A <-> <. B , A >. e. R )
7	4, 5, 6	3bitr3g 211	1 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. `' R <-> <. B , A >. e. R ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   e. wcel 1393   <.cop 3378   class class class wbr 3764   `'ccnv 4344

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353

This theorem is referenced by:  brcnvg  4516  opelcnv  4517  fvimacnv  5282  cnvf1olem  5845  brtposg  5869  xrlenlt  7084