Theorem iinxsng 4049

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 3978	. 2 ⊢ ∩ 𝑥 ∈ {𝐴}𝐵 = {𝑦 ∣ ∀𝑥 ∈ {𝐴}𝑦 ∈ 𝐵}
2		iinxsng.1	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
3	2	eleq2d 2301	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
4	3	ralsng 3713	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶))
5	4	abbi1dv 2352	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∀𝑥 ∈ {𝐴}𝑦 ∈ 𝐵} = 𝐶)
6	1, 5	eqtrid 2276	1 ⊢ (𝐴 ∈ 𝑉 → ∩ 𝑥 ∈ {𝐴}𝐵 = 𝐶)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2202  {cab 2217  ∀wral 2511  {csn 3673  ∩ ciin 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-sbc 3033  df-sn 3679  df-iin 3978

This theorem is referenced by: (None)