Theorem eliin 4716

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 2867	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
2	1	ralbidv 3168	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
3		df-iin 4714	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶}
4	2, 3	elab2g 3546	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1653   ∈ wcel 2157  ∀wral 3090  ∩ ciin 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2778

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-v 3388  df-iin 4714

This theorem is referenced by:  iinconst  4721  iuniin  4722  iinss1  4724  ssiinf  4760  iinss  4762  iinss2  4763  iinab  4772  iinun2  4777  iundif2  4778  iindif2  4780  iinin2  4781  elriin  4784  iinpw  4809  triin  4961  xpiindi  5462  cnviin  5892  iinpreima  6572  iiner  8058  ixpiin  8175  boxriin  8191  iunocv  20349  hauscmplem  21537  txtube  21771  isfcls  22140  iscmet3  23418  taylfval  24453  iinssiun  29893  fnemeet1  32872  diaglbN  37075  dibglbN  37186  dihglbcpreN  37320  kelac1  38413  eliind  39994  eliuniin  40033  eliin2f  40040  eliinid  40047  eliuniin2  40056  iinssiin  40064  eliind2  40065  iinssf  40079  allbutfi  40354  meaiininclem  41441  hspdifhsp  41571  iinhoiicclem  41628  preimageiingt  41671  preimaleiinlt  41672  smflimlem2  41721  smflimsuplem5  41771  smflimsuplem7  41773