Theorem intprg 3955

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3954. 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 3743	. . . 4 |- ( x = A -> { x , y } = { A , y } )
2	1	inteqd 3927	. . 3 |- ( x = A -> |^| { x , y } = |^| { A , y } )
3		ineq1 3398	. . 3 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
4	2, 3	eqeq12d 2244	. 2 |- ( x = A -> ( |^| { x , y } = ( x i^i y ) <-> |^| { A , y } = ( A i^i y ) ) )
5		preq2 3744	. . . 4 |- ( y = B -> { A , y } = { A , B } )
6	5	inteqd 3927	. . 3 |- ( y = B -> |^| { A , y } = |^| { A , B } )
7		ineq2 3399	. . 3 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
8	6, 7	eqeq12d 2244	. 2 |- ( y = B -> ( |^| { A , y } = ( A i^i y ) <-> |^| { A , B } = ( A i^i B ) ) )
9		vex 2802	. . 3 |- x e. _V
10		vex 2802	. . 3 |- y e. _V
11	9, 10	intpr 3954	. 2 |- |^| { x , y } = ( x i^i y )
12	4, 8, 11	vtocl2g 2865	1 |- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   i^i cin 3196   {cpr 3667   |^|cint 3922

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-un 3201  df-in 3203  df-sn 3672  df-pr 3673  df-int 3923

This theorem is referenced by:  intsng  3956  op1stbg  4567  subrngin  14162  subrgin  14193  lssincl  14334