Theorem opelcnvg 5302

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 4657	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 4656	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 5122	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4994	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 4654	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 4654	. 2 |- ( B R A <-> <. B , A >. e. R )
7	4, 5, 6	3bitr3g 302	1 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. `' R <-> <. B , A >. e. R ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   <.cop 4183   class class class wbr 4653   `'ccnv 5113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122

This theorem is referenced by:  brcnvg  5303  opelcnv  5304  elpredim  5692  fvimacnv  6332  brtpos  7361  xrlenlt  10103  brcolinear2  32165