Theorem iinconst 4941

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 4936	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
2	1	elv 3443	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3		r19.3rzv 4435	. . 3 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
4	2, 3	bitr4id 290	. 2 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ 𝐵))
5	4	eqrdv 2734	1 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2104   ≠ wne 2940  ∀wral 3061  Vcvv 3437  ∅c0 4262  ∩ ciin 4932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-v 3439  df-dif 3895  df-nul 4263  df-iin 4934

This theorem is referenced by:  iin0  5293  ptbasfi  22781