Theorem opelcnvg 4859

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 4049	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 4048	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 4684	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4316	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 4046	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 4046	. 2 |- ( B R A <-> <. B , A >. e. R )
7	4, 5, 6	3bitr3g 222	1 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. `' R <-> <. B , A >. e. R ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2176   <.cop 3636   class class class wbr 4045   `'ccnv 4675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-cnv 4684

This theorem is referenced by:  brcnvg  4860  opelcnv  4861  fvimacnv  5697  cnvf1olem  6312  brtposg  6342  xrlenlt  8139