Theorem eliin 4910

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 2900	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
2	1	ralbidv 3197	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
3		df-iin 4908	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶}
4	2, 3	elab2g 3659	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∩ ciin 4906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-iin 4908

This theorem is referenced by:  iinconst  4915  iuniin  4917  iinssiun  4918  iinss1  4920  ssiinf  4964  iinss  4966  iinss2  4967  iinab  4976  iinun2  4981  iundif2  4982  iindif1  4983  iindif2  4985  iinin2  4986  elriin  4989  iinpw  5014  triin  5173  xpiindi  5692  cnviin  6123  iinpreima  6823  iiner  8355  ixpiin  8474  boxriin  8490  iunocv  20808  hauscmplem  21997  txtube  22231  isfcls  22600  iscmet3  23879  taylfval  24933  fnemeet1  33721  diaglbN  38223  dibglbN  38334  dihglbcpreN  38468  kelac1  39755  eliind  41423  eliuniin  41455  eliin2f  41460  eliinid  41467  eliuniin2  41476  iinssiin  41485  eliind2  41486  iinssf  41497  allbutfi  41755  meaiininclem  42858  hspdifhsp  42988  iinhoiicclem  43045  preimageiingt  43088  preimaleiinlt  43089  smflimlem2  43138  smflimsuplem5  43188  smflimsuplem7  43190