Theorem opelcnvg 4679

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 3899	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 3898	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 4507	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4151	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 3896	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 3896	. 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 1463   <.cop 3496   class class class wbr 3895   `'ccnv 4498

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-cnv 4507

This theorem is referenced by:  brcnvg  4680  opelcnv  4681  fvimacnv  5489  cnvf1olem  6075  brtposg  6105  xrlenlt  7753