Theorem iinxprg 4601

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 2687	. . . 4 |- ( x = A -> ( y e. C <-> y e. D ) )
3		iinxprg.2	. . . . 5 |- ( x = B -> C = E )
4	3	eleq2d 2687	. . . 4 |- ( x = B -> ( y e. C <-> y e. E ) )
5	2, 4	ralprg 4234	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } y e. C <-> ( y e. D /\ y e. E ) ) )
6	5	abbidv 2741	. 2 |- ( ( A e. V /\ B e. W ) -> { y | A. x e. { A , B } y e. C } = { y | ( y e. D /\ y e. E ) } )
7		df-iin 4523	. 2 |- |^|_ x e. { A , B } C = { y | A. x e. { A , B } y e. C }
8		df-in 3581	. 2 |- ( D i^i E ) = { y | ( y e. D /\ y e. E ) }
9	6, 7, 8	3eqtr4g 2681	1 |- ( ( A e. V /\ B e. W ) -> |^|_ x e. { A , B } C = ( D i^i E ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   i^i cin 3573   {cpr 4179   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-sbc 3436  df-un 3579  df-in 3581  df-sn 4178  df-pr 4180  df-iin 4523

This theorem is referenced by:  pmapmeet  35059  diameetN  36345  dihmeetlem2N  36588  dihmeetcN  36591  dihmeet  36632