Theorem opelcnvg 4902

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 4087	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 4086	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 4727	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4357	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 4084	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 4084	. 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 2200   <.cop 3669   class class class wbr 4083   `'ccnv 4718

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-cnv 4727

This theorem is referenced by:  brcnvg  4903  opelcnv  4904  fvimacnv  5750  cnvf1olem  6370  brtposg  6400  xrlenlt  8211