Theorem opelcnvg 4533

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 3789	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 3788	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 4371	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4024	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 3786	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 3786	. 2 |- ( B R A <-> <. B , A >. e. R )
7	4, 5, 6	3bitr3g 220	1 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. `' R <-> <. B , A >. e. R ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1433   <.cop 3401   class class class wbr 3785   `'ccnv 4362

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-cnv 4371

This theorem is referenced by:  brcnvg  4534  opelcnv  4535  fvimacnv  5303  cnvf1olem  5865  brtposg  5892  xrlenlt  7177