Theorem intprg 3998

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3997. 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 3784	. . . 4 |- ( x = A -> { x , y } = { A , y } )
2	1	inteqd 3970	. . 3 |- ( x = A -> |^| { x , y } = |^| { A , y } )
3		ineq1 3425	. . 3 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
4	2, 3	eqeq12d 2253	. 2 |- ( x = A -> ( |^| { x , y } = ( x i^i y ) <-> |^| { A , y } = ( A i^i y ) ) )
5		preq2 3785	. . . 4 |- ( y = B -> { A , y } = { A , B } )
6	5	inteqd 3970	. . 3 |- ( y = B -> |^| { A , y } = |^| { A , B } )
7		ineq2 3426	. . 3 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
8	6, 7	eqeq12d 2253	. 2 |- ( y = B -> ( |^| { A , y } = ( A i^i y ) <-> |^| { A , B } = ( A i^i B ) ) )
9		vex 2824	. . 3 |- x e. _V
10		vex 2824	. . 3 |- y e. _V
11	9, 10	intpr 3997	. 2 |- |^| { x , y } = ( x i^i y )
12	4, 8, 11	vtocl2g 2887	1 |- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   i^i cin 3219   {cpr 3706   |^|cint 3965

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-in 3226  df-sn 3711  df-pr 3712  df-int 3966

This theorem is referenced by:  intsng  3999  op1stbg  4620  subrngin  14494  subrgin  14525  lssincl  14694