Theorem iunxsn 5050

Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.)

Hypotheses

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunxsn

Step	Hyp	Ref	Expression
1		iunxsn.1	. 2 ⊢ 𝐴 ∈ V
2		iunxsn.2	. . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
3	2	iunxsng 5049	. 2 ⊢ (𝐴 ∈ V → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶)
4	1, 3	ax-mp 5	1 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3444  {csn 4585  ∪ ciun 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-v 3446  df-sn 4586  df-iun 4953

This theorem is referenced by:  iunsuc  6407  funopsn  7102  fparlem3  8070  fparlem4  8071  iunfi  9270  kmlem11  10092  ackbij1lem8  10157  dfid6  14971  fsum2dlem  15713  fsumiun  15764  fprod2dlem  15923  prmreclem4  16867  fiuncmp  23325  ovolfiniun  25436  finiunmbl  25479  volfiniun  25482  voliunlem1  25485  iuninc  32540  cvmliftlem10  35275  mrsubvrs  35503  dfrcl4  43659  iunrelexp0  43685  corclrcl  43690  cotrcltrcl  43708  trclfvdecomr  43711  dfrtrcl4  43721  corcltrcl  43722  cotrclrcl  43725  imaf1hom  49091