Theorem intprg 3799

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3798. 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 3595	. . . 4 |- ( x = A -> { x , y } = { A , y } )
2	1	inteqd 3771	. . 3 |- ( x = A -> |^| { x , y } = |^| { A , y } )
3		ineq1 3265	. . 3 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
4	2, 3	eqeq12d 2152	. 2 |- ( x = A -> ( |^| { x , y } = ( x i^i y ) <-> |^| { A , y } = ( A i^i y ) ) )
5		preq2 3596	. . . 4 |- ( y = B -> { A , y } = { A , B } )
6	5	inteqd 3771	. . 3 |- ( y = B -> |^| { A , y } = |^| { A , B } )
7		ineq2 3266	. . 3 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
8	6, 7	eqeq12d 2152	. 2 |- ( y = B -> ( |^| { A , y } = ( A i^i y ) <-> |^| { A , B } = ( A i^i B ) ) )
9		vex 2684	. . 3 |- x e. _V
10		vex 2684	. . 3 |- y e. _V
11	9, 10	intpr 3798	. 2 |- |^| { x , y } = ( x i^i y )
12	4, 8, 11	vtocl2g 2745	1 |- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   i^i cin 3065   {cpr 3523   |^|cint 3766

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-un 3070  df-in 3072  df-sn 3528  df-pr 3529  df-int 3767

This theorem is referenced by:  intsng  3800  op1stbg  4395