Theorem opelcnvg 5011

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 4176	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 4175	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 4845	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4434	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 4173	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 4173	. 2 |- ( B R A <-> <. B , A >. e. R )
7	4, 5, 6	3bitr3g 279	1 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. `' R <-> <. B , A >. e. R ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   e. wcel 1721   <.cop 3777   class class class wbr 4172   `'ccnv 4836

This theorem is referenced by:  brcnvg  5012  opelcnv  5013  fvimacnv  5804  brtpos  6447  xrlenlt  9099  elpredim  25390  brcolinear2  25896

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-cnv 4845