Theorem intprg 3669

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3668. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)

Assertion

Ref	Expression
intprg	|- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) )

Proof of Theorem intprg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		preq1 3469	. . . 4 |- ( x = A -> { x , y } = { A , y } )
2	1	inteqd 3641	. . 3 |- ( x = A -> |^| { x , y } = |^| { A , y } )
3		ineq1 3160	. . 3 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
4	2, 3	eqeq12d 2095	. 2 |- ( x = A -> ( |^| { x , y } = ( x i^i y ) <-> |^| { A , y } = ( A i^i y ) ) )
5		preq2 3470	. . . 4 |- ( y = B -> { A , y } = { A , B } )
6	5	inteqd 3641	. . 3 |- ( y = B -> |^| { A , y } = |^| { A , B } )
7		ineq2 3161	. . 3 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
8	6, 7	eqeq12d 2095	. 2 |- ( y = B -> ( |^| { A , y } = ( A i^i y ) <-> |^| { A , B } = ( A i^i B ) ) )
9		vex 2604	. . 3 |- x e. _V
10		vex 2604	. . 3 |- y e. _V
11	9, 10	intpr 3668	. 2 |- |^| { x , y } = ( x i^i y )
12	4, 8, 11	vtocl2g 2662	1 |- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   i^i cin 2972   {cpr 3399   |^|cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-in 2979  df-sn 3404  df-pr 3405  df-int 3637

This theorem is referenced by:  intsng  3670  op1stbg  4228