Theorem opelcnvg 4791

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 3993	. . 3 |- ( x = A -> ( y R x <-> y R A ) )
2		breq1 3992	. . 3 |- ( y = B -> ( y R A <-> B R A ) )
3		df-cnv 4619	. . 3 |- `' R = { <. x , y >. | y R x }
4	1, 2, 3	brabg 4254	. 2 |- ( ( A e. C /\ B e. D ) -> ( A `' R B <-> B R A ) )
5		df-br 3990	. 2 |- ( A `' R B <-> <. A , B >. e. `' R )
6		df-br 3990	. 2 |- ( B R A <-> <. B , A >. e. R )
7	4, 5, 6	3bitr3g 221	1 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. `' R <-> <. B , A >. e. R ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2141   <.cop 3586   class class class wbr 3989   `'ccnv 4610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619

This theorem is referenced by:  brcnvg  4792  opelcnv  4793  fvimacnv  5611  cnvf1olem  6203  brtposg  6233  xrlenlt  7984