Theorem opelcnvg 4940

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 4118	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 4117	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 4762	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4392	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 4115	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 4115	. 2 |- ( B R A <-> <. B , A >. e. R )
7	4, 5, 6	3bitr3g 222	1 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. `' R <-> <. B , A >. e. R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2205   <.cop 3697   class class class wbr 4114   `'ccnv 4753

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-cnv 4762

This theorem is referenced by:  brcnvg  4941  opelcnv  4942  fvimacnv  5798  cnvf1olem  6433  brtposg  6498  xrlenlt  8354