Theorem dfopg 3897

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 2833	. 2 |- ( A e. V -> A e. _V )
2		elex 2833	. 2 |- ( B e. W -> B e. _V )
3		df-3an 1011	. . . . . 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 932	. . . . 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 2358	. . . 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 2361	. . . 4 |- { x | x e. { { A } , { A , B } } } = { { A } , { A , B } }
7		df-op 3714	. . . . 5 |- <. A , B >. = { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) }
8	7	eqcomi 2242	. . . 4 |- { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) } = <. A , B >.
9	5, 6, 8	3eqtr3g 2294	. . 3 |- ( ( A e. _V /\ B e. _V ) -> { { A } , { A , B } } = <. A , B >. )
10	9	eqcomd 2244	. 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 1009   = wceq 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   {csn 3705   {cpr 3706   <.cop 3708

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823  df-op 3714

This theorem is referenced by:  dfop  3898  opexg  4363  opth1  4371  opth  4372  0nelop  4383  op1stbg  4620