Theorem opelcnvg 5180

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 4451	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 4450	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 5007	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4766	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 4448	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 4448	. 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 1767   <.cop 4033   class class class wbr 4447   `'ccnv 4998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-cnv 5007

This theorem is referenced by:  brcnvg  5181  opelcnv  5182  fvimacnv  5994  brtpos  6961  xrlenlt  9648  elpredim  28830  brcolinear2  29282