Theorem iunxsn 5047

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 5046	. 2 ⊢ (𝐴 ∈ V → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶)
4	1, 3	ax-mp 5	1 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3441  {csn 4581  ∪ ciun 4947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-v 3443  df-sn 4582  df-iun 4949

This theorem is referenced by:  iunsuc  6405  funopsn  7095  fparlem3  8058  fparlem4  8059  iunfi  9247  kmlem11  10075  ackbij1lem8  10140  dfid6  14955  fsum2dlem  15697  fsumiun  15748  fprod2dlem  15907  prmreclem4  16851  fiuncmp  23352  ovolfiniun  25462  finiunmbl  25505  volfiniun  25508  voliunlem1  25511  iuninc  32617  cvmliftlem10  35469  mrsubvrs  35697  dfrcl4  43953  iunrelexp0  43979  corclrcl  43984  cotrcltrcl  44002  trclfvdecomr  44005  dfrtrcl4  44015  corcltrcl  44016  cotrclrcl  44019  imaf1hom  49389