Theorem eliin 4275

Description: Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)

Assertion

Ref	Expression
eliin	|- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   C( x)   V( x)

Proof of Theorem eliin

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2537	. . 3 |- ( y = A -> ( y e. C <-> A e. C ) )
2	1	ralbidv 2829	. 2 |- ( y = A -> ( A. x e. B y e. C <-> A. x e. B A e. C ) )
3		df-iin 4272	. 2 |- |^|_ x e. B C = { y | A. x e. B y e. C }
4	2, 3	elab2g 3175	1 |- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )

Syntax hints:   -> wi 4   <-> wb 189   = wceq 1452   e. wcel 1904   A.wral 2756   |^|_ciin 4270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-v 3033  df-iin 4272

This theorem is referenced by:  iinconst  4279  iuniin  4280  iinss1  4282  ssiinf  4318  iinss  4320  iinss2  4321  iinab  4330  iinun2  4335  iundif2  4336  iindif2  4338  iinin2  4339  elriin  4342  iinpw  4363  xpiindi  4975  cnviin  5380  iinpreima  6025  iiner  7453  ixpiin  7566  boxriin  7582  iunocv  19321  hauscmplem  20498  txtube  20732  isfcls  21102  iscmet3  22341  taylfval  23393  fnemeet1  31093  diaglbN  34694  dibglbN  34805  dihglbcpreN  34939  kelac1  35992  eliind  37474  hspdifhsp  38556