Theorem intprg 3982

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3981. 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 3768	. . . 4 |- ( x = A -> { x , y } = { A , y } )
2	1	inteqd 3954	. . 3 |- ( x = A -> |^| { x , y } = |^| { A , y } )
3		ineq1 3415	. . 3 |- ( x = A -> ( x i^i y ) = ( A i^i y ) )
4	2, 3	eqeq12d 2247	. 2 |- ( x = A -> ( |^| { x , y } = ( x i^i y ) <-> |^| { A , y } = ( A i^i y ) ) )
5		preq2 3769	. . . 4 |- ( y = B -> { A , y } = { A , B } )
6	5	inteqd 3954	. . 3 |- ( y = B -> |^| { A , y } = |^| { A , B } )
7		ineq2 3416	. . 3 |- ( y = B -> ( A i^i y ) = ( A i^i B ) )
8	6, 7	eqeq12d 2247	. 2 |- ( y = B -> ( |^| { A , y } = ( A i^i y ) <-> |^| { A , B } = ( A i^i B ) ) )
9		vex 2816	. . 3 |- x e. _V
10		vex 2816	. . 3 |- y e. _V
11	9, 10	intpr 3981	. 2 |- |^| { x , y } = ( x i^i y )
12	4, 8, 11	vtocl2g 2879	1 |- ( ( A e. V /\ B e. W ) -> |^| { A , B } = ( A i^i B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   i^i cin 3210   {cpr 3690   |^|cint 3949

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815  df-un 3215  df-in 3217  df-sn 3695  df-pr 3696  df-int 3950

This theorem is referenced by:  intsng  3983  op1stbg  4600  subrngin  14358  subrgin  14389  lssincl  14533