Theorem opelcnvg 4719

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 3933	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 3932	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 4547	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4191	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 3930	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 3930	. 2 |- ( B R A <-> <. B , A >. e. R )
7	4, 5, 6	3bitr3g 221	1 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. `' R <-> <. B , A >. e. R ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1480   <.cop 3530   class class class wbr 3929   `'ccnv 4538

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547

This theorem is referenced by:  brcnvg  4720  opelcnv  4721  fvimacnv  5535  cnvf1olem  6121  brtposg  6151  xrlenlt  7829