Theorem opelcnvg 5124

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 4398	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 4397	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 4950	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4708	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 4395	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 4395	. 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 367   e. wcel 1842   <.cop 3977   class class class wbr 4394   `'ccnv 4941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-cnv 4950

This theorem is referenced by:  brcnvg  5125  opelcnv  5126  fvimacnv  5936  brtpos  6921  xrlenlt  9602  elpredim  29929  brcolinear2  30369