Theorem eliin 4324

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 2532	. . 3 |- ( y = A -> ( y e. C <-> A e. C ) )
2	1	ralbidv 2896	. 2 |- ( y = A -> ( A. x e. B y e. C <-> A. x e. B A e. C ) )
3		df-iin 4321	. 2 |- |^|_ x e. B C = { y | A. x e. B y e. C }
4	2, 3	elab2g 3245	1 |- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1374   e. wcel 1762   A.wral 2807   |^|_ciin 4319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-v 3108  df-iin 4321

This theorem is referenced by:  iinconst  4328  iuniin  4329  iinss1  4331  ssiinf  4367  iinss  4369  iinss2  4370  iinab  4379  iinun2  4384  iundif2  4385  iindif2  4387  iinin2  4388  elriin  4391  iinpw  4407  xpiindi  5129  cnviin  5535  iinpreima  6002  iiner  7373  ixpiin  7485  boxriin  7501  iunocv  18472  hauscmplem  19665  txtube  19869  isfcls  20238  iscmet3  21460  taylfval  22481  fnemeet1  29774  kelac1  30602  diaglbN  35727  dibglbN  35838  dihglbcpreN  35972