Theorem eliin 4955

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 2848	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
2	1	ralbidv 3185	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
3		df-iin 4953	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶}
4	2, 3	elab2g 3633	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∩ ciin 4951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-iin 4953

This theorem is used by:  iinconst  4961  iuniin  4963  iinssiun  4964  iinss1  4966  ssiinf  5012  iinss  5014  iinss2  5015  iinab  5025  iinun2  5030  iundif2  5031  iindif1  5034  iindif2  5036  iinin2  5037  elriin  5040  iinpw  5065  triin  5228  xpiindi  5811  cnviin  6289  iinpreima  7069  iiner  8813  ixpiin  8955  boxriin  8971  iunocv  21992  hauscmplem  23729  txtube  23964  isfcls  24333  iscmet3  25619  taylfval  26691  suppgsumssiun  33679  zarclsiin  34549  fnemeet1  37246  diaglbN  42205  dibglbN  42316  dihglbcpreN  42450  kelac1  44158  eliind  46181  eliuniin  46207  eliin2f  46212  eliinid  46219  eliuniin2  46228  iinssiin  46237  eliind2  46238  iinssf  46246  iindif2f  46268  allbutfi  46497  meaiininclem  47589  hspdifhsp  47719  iinhoiicclem  47776  preimageiingt  47823  preimaleiinlt  47824  smflimlem2  47875  smflimsuplem5  47927  smflimsuplem7  47929  iineq0  50024  iinxp  50035  iinfsubc  50265