Theorem intprg 3873

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3872. 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 3666	. . . 4 |- ( x = A -> { x , y } = { A , y } )
2	1	inteqd 3845	. . 3 |- ( x = A -> |^| { x , y } = |^| { A , y } )
3		ineq1 3327	. . 3 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
4	2, 3	eqeq12d 2190	. 2 |- ( x = A -> ( |^| { x , y } = ( x i^i y ) <-> |^| { A , y } = ( A i^i y ) ) )
5		preq2 3667	. . . 4 |- ( y = B -> { A , y } = { A , B } )
6	5	inteqd 3845	. . 3 |- ( y = B -> |^| { A , y } = |^| { A , B } )
7		ineq2 3328	. . 3 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
8	6, 7	eqeq12d 2190	. 2 |- ( y = B -> ( |^| { A , y } = ( A i^i y ) <-> |^| { A , B } = ( A i^i B ) ) )
9		vex 2738	. . 3 |- x e. _V
10		vex 2738	. . 3 |- y e. _V
11	9, 10	intpr 3872	. 2 |- |^| { x , y } = ( x i^i y )
12	4, 8, 11	vtocl2g 2799	1 |- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2146   i^i cin 3126   {cpr 3590   |^|cint 3840

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-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-v 2737  df-un 3131  df-in 3133  df-sn 3595  df-pr 3596  df-int 3841

This theorem is referenced by:  intsng  3874  op1stbg  4473