Theorem intprg 3639

Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3638. 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 3438	. . . 4 |- (x = A -> {x , y } = {A , y })
2	1	inteqd 3611	. . 3 |- (x = A -> |^| {x , y } = |^| {A , y })
3		ineq1 3125	. . 3 |- (x = A -> (x i^i y) = (A i^i y))
4	2, 3	eqeq12d 2051	. 2 |- (x = A -> ( |^| {x , y } = (x i^i y) <-> |^| {A , y } = (A i^i y)))
5		preq2 3439	. . . 4 |- (y = B -> {A , y } = {A , B })
6	5	inteqd 3611	. . 3 |- (y = B -> |^| {A , y } = |^| {A , B })
7		ineq2 3126	. . 3 |- (y = B -> (A i^i y) = (A i^i B))
8	6, 7	eqeq12d 2051	. 2 |- (y = B -> ( |^| {A , y } = (A i^i y) <-> |^| {A , B } = (A i^i B)))
9		vex 2554	. . 3 |- x e. _V
10		vex 2554	. . 3 |- y e. _V
11	9, 10	intpr 3638	. 2 |- |^| {x , y } = (x i^i y)
12	4, 8, 11	vtocl2g 2611	1 |- ((A e. V /\ B e. W) -> |^| {A , B } = (A i^i B))

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242   e. wcel 1390   i^i cin 2910   {cpr 3368   |^|cint 3606

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-sn 3373  df-pr 3374  df-int 3607

This theorem is referenced by:  intsng  3640  op1stbg  4176