Theorem intprg 3920

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3919. 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 3711	. . . 4 |- ( x = A -> { x , y } = { A , y } )
2	1	inteqd 3892	. . 3 |- ( x = A -> |^| { x , y } = |^| { A , y } )
3		ineq1 3368	. . 3 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
4	2, 3	eqeq12d 2221	. 2 |- ( x = A -> ( |^| { x , y } = ( x i^i y ) <-> |^| { A , y } = ( A i^i y ) ) )
5		preq2 3712	. . . 4 |- ( y = B -> { A , y } = { A , B } )
6	5	inteqd 3892	. . 3 |- ( y = B -> |^| { A , y } = |^| { A , B } )
7		ineq2 3369	. . 3 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
8	6, 7	eqeq12d 2221	. 2 |- ( y = B -> ( |^| { A , y } = ( A i^i y ) <-> |^| { A , B } = ( A i^i B ) ) )
9		vex 2776	. . 3 |- x e. _V
10		vex 2776	. . 3 |- y e. _V
11	9, 10	intpr 3919	. 2 |- |^| { x , y } = ( x i^i y )
12	4, 8, 11	vtocl2g 2838	1 |- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   i^i cin 3166   {cpr 3635   |^|cint 3887

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-un 3171  df-in 3173  df-sn 3640  df-pr 3641  df-int 3888

This theorem is referenced by:  intsng  3921  op1stbg  4530  subrngin  14019  subrgin  14050  lssincl  14191