Theorem opelcnvg 4910

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 4092	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 4091	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 4733	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4363	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 4089	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 4089	. 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 2202   <.cop 3672   class class class wbr 4088   `'ccnv 4724

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-cnv 4733

This theorem is referenced by:  brcnvg  4911  opelcnv  4912  fvimacnv  5762  cnvf1olem  6388  brtposg  6419  xrlenlt  8243