Theorem dfopg 3806

Description: Value of the ordered pair when the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
dfopg	|- ( ( A e. V /\ B e. W ) -> <. A , B >. = { { A } , { A , B } } )

Proof of Theorem dfopg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2774	. 2 |- ( A e. V -> A e. _V )
2		elex 2774	. 2 |- ( B e. W -> B e. _V )
3		df-3an 982	. . . . . 6 |- ( ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) <-> ( ( A e. _V /\ B e. _V ) /\ x e. { { A } , { A , B } } ) )
4	3	baibr 921	. . . . 5 |- ( ( A e. _V /\ B e. _V ) -> ( x e. { { A } , { A , B } } <-> ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) ) )
5	4	abbidv 2314	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> { x | x e. { { A } , { A , B } } } = { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) } )
6		abid2 2317	. . . 4 |- { x | x e. { { A } , { A , B } } } = { { A } , { A , B } }
7		df-op 3631	. . . . 5 |- <. A , B >. = { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) }
8	7	eqcomi 2200	. . . 4 |- { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) } = <. A , B >.
9	5, 6, 8	3eqtr3g 2252	. . 3 |- ( ( A e. _V /\ B e. _V ) -> { { A } , { A , B } } = <. A , B >. )
10	9	eqcomd 2202	. 2 |- ( ( A e. _V /\ B e. _V ) -> <. A , B >. = { { A } , { A , B } } )
11	1, 2, 10	syl2an 289	1 |- ( ( A e. V /\ B e. W ) -> <. A , B >. = { { A } , { A , B } } )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   {cab 2182   _Vcvv 2763   {csn 3622   {cpr 3623   <.cop 3625

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-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765  df-op 3631

This theorem is referenced by:  dfop  3807  opexg  4261  opth1  4269  opth  4270  0nelop  4281  op1stbg  4514