Theorem eliin 4174

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 2501	. . 3 |- ( y = A -> ( y e. C <-> A e. C ) )
2	1	ralbidv 2733	. 2 |- ( y = A -> ( A. x e. B y e. C <-> A. x e. B A e. C ) )
3		df-iin 4172	. 2 |- |^|_ x e. B C = { y | A. x e. B y e. C }
4	2, 3	elab2g 3106	1 |- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1369   e. wcel 1756   A.wral 2713   |^|_ciin 4170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-v 2972  df-iin 4172

This theorem is referenced by:  iinconst  4178  iuniin  4179  iinss1  4181  ssiinf  4217  iinss  4219  iinss2  4220  iinab  4229  iinun2  4234  iundif2  4235  iindif2  4237  iinin2  4238  elriin  4241  iinpw  4257  xpiindi  4973  cnviin  5372  iinpreima  5831  iiner  7170  ixpiin  7287  boxriin  7303  iunocv  18104  hauscmplem  19007  txtube  19211  isfcls  19580  iscmet3  20802  taylfval  21822  fnemeet1  28584  kelac1  29413  diaglbN  34697  dibglbN  34808  dihglbcpreN  34942