Theorem iinxprg 4043

Description: Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)

Hypotheses

Ref	Expression
iinxprg.1	|- (x = A -> C = D)
iinxprg.2	|- (x = B -> C = E)

Assertion

Ref	Expression
iinxprg	|- ((A e. V /\ B e. W) -> |^|_x e. {A, B}C = (D i^i E))

Distinct variable groups:   x,A   x,B   x,D   x,E

Allowed substitution hints:   C(x)   V(x)   W(x)

Proof of Theorem iinxprg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iinxprg.1	. . . . 5 |- (x = A -> C = D)
2	1	eleq2d 2420	. . . 4 |- (x = A -> (y e. C <-> y e. D))
3		iinxprg.2	. . . . 5 |- (x = B -> C = E)
4	3	eleq2d 2420	. . . 4 |- (x = B -> (y e. C <-> y e. E))
5	2, 4	ralprg 3775	. . 3 |- ((A e. V /\ B e. W) -> (A.x e. {A, B}y e. C <-> (y e. D /\ y e. E)))
6		vex 2862	. . . 4 |- y e. _V
7		eliin 3974	. . . 4 |- (y e. _V -> (y e. |^|_x e. {A, B}C <-> A.x e. {A, B}y e. C))
8	6, 7	ax-mp 8	. . 3 |- (y e. |^|_x e. {A, B}C <-> A.x e. {A, B}y e. C)
9		elin 3219	. . 3 |- (y e. (D i^i E) <-> (y e. D /\ y e. E))
10	5, 8, 9	3bitr4g 279	. 2 |- ((A e. V /\ B e. W) -> (y e. |^|_x e. {A, B}C <-> y e. (D i^i E)))
11	10	eqrdv 2351	1 |- ((A e. V /\ B e. W) -> |^|_x e. {A, B}C = (D i^i E))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2614  _Vcvv 2859   i^i cin 3208  {cpr 3738  |^|_ciin 3970

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-sn 3741  df-pr 3742  df-iin 3972

This theorem is referenced by: (None)