Theorem intprg 3907

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3906. 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 3699	. . . 4 |- ( x = A -> { x , y } = { A , y } )
2	1	inteqd 3879	. . 3 |- ( x = A -> |^| { x , y } = |^| { A , y } )
3		ineq1 3357	. . 3 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
4	2, 3	eqeq12d 2211	. 2 |- ( x = A -> ( |^| { x , y } = ( x i^i y ) <-> |^| { A , y } = ( A i^i y ) ) )
5		preq2 3700	. . . 4 |- ( y = B -> { A , y } = { A , B } )
6	5	inteqd 3879	. . 3 |- ( y = B -> |^| { A , y } = |^| { A , B } )
7		ineq2 3358	. . 3 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
8	6, 7	eqeq12d 2211	. 2 |- ( y = B -> ( |^| { A , y } = ( A i^i y ) <-> |^| { A , B } = ( A i^i B ) ) )
9		vex 2766	. . 3 |- x e. _V
10		vex 2766	. . 3 |- y e. _V
11	9, 10	intpr 3906	. 2 |- |^| { x , y } = ( x i^i y )
12	4, 8, 11	vtocl2g 2828	1 |- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   i^i cin 3156   {cpr 3623   |^|cint 3874

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-un 3161  df-in 3163  df-sn 3628  df-pr 3629  df-int 3875

This theorem is referenced by:  intsng  3908  op1stbg  4514  subrngin  13769  subrgin  13800  lssincl  13941