Theorem opelcnvg 4955

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 4129	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 4128	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 4777	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4406	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 4126	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 4126	. 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 2209   <.cop 3708   class class class wbr 4125   `'ccnv 4768

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-cnv 4777

This theorem is referenced by:  brcnvg  4956  opelcnv  4957  fvimacnv  5815  cnvf1olem  6450  brtposg  6515  xrlenlt  8380