Theorem eliin 5000

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

Assertion

Ref	Expression
eliin	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 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 2826	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
2	1	ralbidv 3175	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
3		df-iin 4998	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶}
4	2, 3	elab2g 3682	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ∩ ciin 4996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-iin 4998

This theorem is referenced by:  iinconst  5006  iuniin  5008  iinssiun  5009  iinss1  5011  ssiinf  5058  iinss  5060  iinss2  5061  iinab  5072  iinun2  5077  iundif2  5078  iindif1  5079  iindif2  5081  iinin2  5082  elriin  5085  iinpw  5110  triin  5281  xpiindi  5848  cnviin  6307  iinpreima  7088  iiner  8827  ixpiin  8962  boxriin  8978  iunocv  21716  hauscmplem  23429  txtube  23663  isfcls  24032  iscmet3  25340  taylfval  26414  zarclsiin  33831  fnemeet1  36348  diaglbN  41037  dibglbN  41148  dihglbcpreN  41282  kelac1  43051  eliind  45010  eliuniin  45038  eliin2f  45043  eliinid  45050  eliuniin2  45059  iinssiin  45068  eliind2  45069  iinssf  45077  iindif2f  45102  allbutfi  45342  meaiininclem  46441  hspdifhsp  46571  iinhoiicclem  46628  preimageiingt  46675  preimaleiinlt  46676  smflimlem2  46727  smflimsuplem5  46779  smflimsuplem7  46781