Theorem opelcnvg 5126

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 4403	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 4402	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 4955	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4715	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 4400	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 4400	. 2 |- ( B R A <-> <. B , A >. e. R )
7	4, 5, 6	3bitr3g 287	1 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. `' R <-> <. B , A >. e. R ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1758   <.cop 3990   class class class wbr 4399   `'ccnv 4946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-br 4400  df-opab 4458  df-cnv 4955

This theorem is referenced by:  brcnvg  5127  opelcnv  5128  fvimacnv  5926  brtpos  6863  xrlenlt  9552  elpredim  27780  brcolinear2  28232