Theorem eliin 4886

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 2877	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
2	1	ralbidv 3162	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
3		df-iin 4884	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶}
4	2, 3	elab2g 3616	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∩ ciin 4882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-iin 4884

This theorem is referenced by:  iinconst  4891  iuniin  4893  iinssiun  4894  iinss1  4896  ssiinf  4941  iinss  4943  iinss2  4944  iinab  4953  iinun2  4958  iundif2  4959  iindif1  4960  iindif2  4962  iinin2  4963  elriin  4966  iinpw  4991  triin  5151  xpiindi  5670  cnviin  6105  iinpreima  6814  iiner  8352  ixpiin  8471  boxriin  8487  iunocv  20370  hauscmplem  22011  txtube  22245  isfcls  22614  iscmet3  23897  taylfval  24954  zarclsiin  31224  fnemeet1  33827  diaglbN  38351  dibglbN  38462  dihglbcpreN  38596  kelac1  40007  eliind  41705  eliuniin  41735  eliin2f  41740  eliinid  41747  eliuniin2  41755  iinssiin  41764  eliind2  41765  iinssf  41775  allbutfi  42029  meaiininclem  43125  hspdifhsp  43255  iinhoiicclem  43312  preimageiingt  43355  preimaleiinlt  43356  smflimlem2  43405  smflimsuplem5  43455  smflimsuplem7  43457