Theorem opelcnvg 4784

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 3986	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 3985	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 4612	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4247	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 3983	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 3983	. 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 2136   <.cop 3579   class class class wbr 3982   `'ccnv 4603

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-cnv 4612

This theorem is referenced by:  brcnvg  4785  opelcnv  4786  fvimacnv  5600  cnvf1olem  6192  brtposg  6222  xrlenlt  7963