Theorem iinxprg 3973

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 2257	. . . 4 |- ( x = A -> ( y e. C <-> y e. D ) )
3		iinxprg.2	. . . . 5 |- ( x = B -> C = E )
4	3	eleq2d 2257	. . . 4 |- ( x = B -> ( y e. C <-> y e. E ) )
5	2, 4	ralprg 3655	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } y e. C <-> ( y e. D /\ y e. E ) ) )
6	5	abbidv 2305	. 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 3901	. 2 |- |^|_ x e. { A , B } C = { y | A. x e. { A , B } y e. C }
8		df-in 3147	. 2 |- ( D i^i E ) = { y | ( y e. D /\ y e. E ) }
9	6, 7, 8	3eqtr4g 2245	1 |- ( ( A e. V /\ B e. W ) -> |^|_ x e. { A , B } C = ( D i^i E ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2158   {cab 2173   A.wral 2465   i^i cin 3140   {cpr 3605   |^|_ciin 3899

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-sn 3610  df-pr 3611  df-iin 3901

This theorem is referenced by: (None)