Theorem iinxsng 5042

Description: A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Hypothesis

Ref	Expression
iinxsng.1	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
iinxsng	⊢ (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}𝐵 = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem iinxsng

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 4948	. 2 ⊢ ∩ 𝑥 ∈ {𝐴}𝐵 = {𝑦 ∣ ∀𝑥 ∈ {𝐴}𝑦 ∈ 𝐵}
2		iinxsng.1	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
3	2	eleq2d 2821	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
4	3	ralsng 4631	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
5	4	eqabcdv 2869	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∀𝑥 ∈ {𝐴}𝑦 ∈ 𝐵} = 𝐶)
6	1, 5	eqtrid 2782	1 ⊢ (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}𝐵 = 𝐶)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {cab 2713  ∀wral 3050  {csn 4579  ∩ ciin 4946

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-v 3441  df-sn 4580  df-iin 4948

This theorem is referenced by:  polatN  40226