Theorem iinxprg 3963

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 2247	. . . 4 |- ( x = A -> ( y e. C <-> y e. D ) )
3		iinxprg.2	. . . . 5 |- ( x = B -> C = E )
4	3	eleq2d 2247	. . . 4 |- ( x = B -> ( y e. C <-> y e. E ) )
5	2, 4	ralprg 3645	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } y e. C <-> ( y e. D /\ y e. E ) ) )
6	5	abbidv 2295	. 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 3891	. 2 |- |^|_ x e. { A , B } C = { y | A. x e. { A , B } y e. C }
8		df-in 3137	. 2 |- ( D i^i E ) = { y | ( y e. D /\ y e. E ) }
9	6, 7, 8	3eqtr4g 2235	1 |- ( ( A e. V /\ B e. W ) -> |^|_ x e. { A , B } C = ( D i^i E ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   i^i cin 3130   {cpr 3595   |^|_ciin 3889

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-sn 3600  df-pr 3601  df-iin 3891

This theorem is referenced by: (None)