Theorem iunxsnf 42636

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 5024	. 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 2101  Ⅎwnfc 2882  Vcvv 3434  {csn 4564  ∪ ciun 4927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-12 2166  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-rex 3069  df-v 3436  df-sbc 3719  df-sn 4565  df-iun 4929

This theorem is referenced by:  fiiuncl  42637  iunp1  42638  sge0iunmptlemfi  43987