Theorem iinxprg 4004

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 2276	. . . 4 |- ( x = A -> ( y e. C <-> y e. D ) )
3		iinxprg.2	. . . . 5 |- ( x = B -> C = E )
4	3	eleq2d 2276	. . . 4 |- ( x = B -> ( y e. C <-> y e. E ) )
5	2, 4	ralprg 3685	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } y e. C <-> ( y e. D /\ y e. E ) ) )
6	5	abbidv 2324	. 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 3932	. 2 |- |^|_ x e. { A , B } C = { y | A. x e. { A , B } y e. C }
8		df-in 3173	. 2 |- ( D i^i E ) = { y | ( y e. D /\ y e. E ) }
9	6, 7, 8	3eqtr4g 2264	1 |- ( ( A e. V /\ B e. W ) -> |^|_ x e. { A , B } C = ( D i^i E ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   {cab 2192   A.wral 2485   i^i cin 3166   {cpr 3635   |^|_ciin 3930

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-sbc 3000  df-un 3171  df-in 3173  df-sn 3640  df-pr 3641  df-iin 3932

This theorem is referenced by: (None)