Theorem iunxsnf 45004

Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
iunxsnf.1	⊢ Ⅎ𝑥𝐶
iunxsnf.2	⊢ 𝐴 ∈ V
iunxsnf.3	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
iunxsnf	⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem iunxsnf

Step	Hyp	Ref	Expression
1		iunxsnf.2	. 2 ⊢ 𝐴 ∈ V
2		iunxsnf.1	. . 3 ⊢ Ⅎ𝑥𝐶
3		iunxsnf.3	. . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
4	2, 3	iunxsngf 5097	. 2 ⊢ (𝐴 ∈ V → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶)
5	1, 4	ax-mp 5	1 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  Ⅎwnfc 2888  Vcvv 3478  {csn 4631  ∪ ciun 4996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rex 3069  df-v 3480  df-sbc 3792  df-sn 4632  df-iun 4998

This theorem is referenced by:  fiiuncl  45005  iunp1  45006  sge0iunmptlemfi  46369