Theorem iinconst 5001

Description: Indexed intersection of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by Mario Carneiro, 6-Feb-2015.)

Assertion

Ref	Expression
iinconst	⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iinconst

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliin 4996	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
2	1	elv 3475	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3		r19.3rzv 4494	. . 3 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
4	2, 3	bitr4id 290	. 2 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ 𝐵))
5	4	eqrdv 2725	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1534   ∈ wcel 2099   ≠ wne 2935  ∀wral 3056  Vcvv 3469  ∅c0 4318  ∩ ciin 4992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2164  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-v 3471  df-dif 3947  df-nul 4319  df-iin 4994

This theorem is referenced by:  iin0  5356  ptbasfi  23472