Theorem opelcnvg 5014

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 4406	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 4405	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 4842	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4720	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 4403	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 4403	. 2 |- ( B R A <-> <. B , A >. e. R )
7	4, 5, 6	3bitr3g 291	1 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. `' R <-> <. B , A >. e. R ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   e. wcel 1887   <.cop 3974   class class class wbr 4402   `'ccnv 4833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-cnv 4842

This theorem is referenced by:  brcnvg  5015  opelcnv  5016  elpredim  5392  fvimacnv  5997  brtpos  6982  xrlenlt  9699  brcolinear2  30825