Theorem intprg 4511

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 4510. 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 4268	. . . 4 |- ( x = A -> { x , y } = { A , y } )
2	1	inteqd 4480	. . 3 |- ( x = A -> |^| { x , y } = |^| { A , y } )
3		ineq1 3807	. . 3 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
4	2, 3	eqeq12d 2637	. 2 |- ( x = A -> ( |^| { x , y } = ( x i^i y ) <-> |^| { A , y } = ( A i^i y ) ) )
5		preq2 4269	. . . 4 |- ( y = B -> { A , y } = { A , B } )
6	5	inteqd 4480	. . 3 |- ( y = B -> |^| { A , y } = |^| { A , B } )
7		ineq2 3808	. . 3 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
8	6, 7	eqeq12d 2637	. 2 |- ( y = B -> ( |^| { A , y } = ( A i^i y ) <-> |^| { A , B } = ( A i^i B ) ) )
9		vex 3203	. . 3 |- x e. _V
10		vex 3203	. . 3 |- y e. _V
11	9, 10	intpr 4510	. 2 |- |^| { x , y } = ( x i^i y )
12	4, 8, 11	vtocl2g 3270	1 |- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   {cpr 4179   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-un 3579  df-in 3581  df-sn 4178  df-pr 4180  df-int 4476

This theorem is referenced by:  intsng  4512  inelfi  8324  mreincl  16259  subrgin  18803  lssincl  18965  incld  20847  difelsiga  30196  inelpisys  30217  inidl  33829