Theorem iinxprg 3947

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 2240	. . . 4 |- ( x = A -> ( y e. C <-> y e. D ) )
3		iinxprg.2	. . . . 5 |- ( x = B -> C = E )
4	3	eleq2d 2240	. . . 4 |- ( x = B -> ( y e. C <-> y e. E ) )
5	2, 4	ralprg 3634	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } y e. C <-> ( y e. D /\ y e. E ) ) )
6	5	abbidv 2288	. 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 3876	. 2 |- |^|_ x e. { A , B } C = { y | A. x e. { A , B } y e. C }
8		df-in 3127	. 2 |- ( D i^i E ) = { y | ( y e. D /\ y e. E ) }
9	6, 7, 8	3eqtr4g 2228	1 |- ( ( A e. V /\ B e. W ) -> |^|_ x e. { A , B } C = ( D i^i E ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   {cab 2156   A.wral 2448   i^i cin 3120   {cpr 3584   |^|_ciin 3874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-sn 3589  df-pr 3590  df-iin 3876

This theorem is referenced by: (None)