Theorem iunxsnf 39233

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	iunxsngf2 39230	. 2 ⊢ (𝐴 ∈ V → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶)
5	1, 4	ax-mp 5	1 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Ⅎwnfc 2751  Vcvv 3200  {csn 4177  ∪ ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-sn 4178  df-iun 4522

This theorem is referenced by:  fiiuncl  39234  iunp1  39235  sge0iunmptlemfi  40630