Theorem intprg 3857

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3856. 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 3653	. . . 4 |- ( x = A -> { x , y } = { A , y } )
2	1	inteqd 3829	. . 3 |- ( x = A -> |^| { x , y } = |^| { A , y } )
3		ineq1 3316	. . 3 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
4	2, 3	eqeq12d 2180	. 2 |- ( x = A -> ( |^| { x , y } = ( x i^i y ) <-> |^| { A , y } = ( A i^i y ) ) )
5		preq2 3654	. . . 4 |- ( y = B -> { A , y } = { A , B } )
6	5	inteqd 3829	. . 3 |- ( y = B -> |^| { A , y } = |^| { A , B } )
7		ineq2 3317	. . 3 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
8	6, 7	eqeq12d 2180	. 2 |- ( y = B -> ( |^| { A , y } = ( A i^i y ) <-> |^| { A , B } = ( A i^i B ) ) )
9		vex 2729	. . 3 |- x e. _V
10		vex 2729	. . 3 |- y e. _V
11	9, 10	intpr 3856	. 2 |- |^| { x , y } = ( x i^i y )
12	4, 8, 11	vtocl2g 2790	1 |- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   i^i cin 3115   {cpr 3577   |^|cint 3824

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-in 3122  df-sn 3582  df-pr 3583  df-int 3825

This theorem is referenced by:  intsng  3858  op1stbg  4457