Theorem iunxsn 5067

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 5066	. 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 2108  Vcvv 3459  {csn 4601  ∪ ciun 4967

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-v 3461  df-sn 4602  df-iun 4969

This theorem is referenced by:  iunsuc  6439  funopsn  7138  fparlem3  8113  fparlem4  8114  iunfi  9355  kmlem11  10175  ackbij1lem8  10240  dfid6  15047  fsum2dlem  15786  fsumiun  15837  fprod2dlem  15996  prmreclem4  16939  fiuncmp  23342  ovolfiniun  25454  finiunmbl  25497  volfiniun  25500  voliunlem1  25503  iuninc  32541  cvmliftlem10  35316  mrsubvrs  35544  dfrcl4  43700  iunrelexp0  43726  corclrcl  43731  cotrcltrcl  43749  trclfvdecomr  43752  dfrtrcl4  43762  corcltrcl  43763  cotrclrcl  43766  imaf1hom  49067