Theorem intprg 3648

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3647. 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 3447	. . . 4 |- ( x = A -> { x , y } = { A , y } )
2	1	inteqd 3620	. . 3 |- ( x = A -> |^| { x , y } = |^| { A , y } )
3		ineq1 3131	. . 3 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
4	2, 3	eqeq12d 2054	. 2 |- ( x = A -> ( |^| { x , y } = ( x i^i y ) <-> |^| { A , y } = ( A i^i y ) ) )
5		preq2 3448	. . . 4 |- ( y = B -> { A , y } = { A , B } )
6	5	inteqd 3620	. . 3 |- ( y = B -> |^| { A , y } = |^| { A , B } )
7		ineq2 3132	. . 3 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
8	6, 7	eqeq12d 2054	. 2 |- ( y = B -> ( |^| { A , y } = ( A i^i y ) <-> |^| { A , B } = ( A i^i B ) ) )
9		vex 2560	. . 3 |- x e. _V
10		vex 2560	. . 3 |- y e. _V
11	9, 10	intpr 3647	. 2 |- |^| { x , y } = ( x i^i y )
12	4, 8, 11	vtocl2g 2617	1 |- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   i^i cin 2916   {cpr 3376   |^|cint 3615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-sn 3381  df-pr 3382  df-int 3616

This theorem is referenced by:  intsng  3649  op1stbg  4210