Theorem iunxsn 5034

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 5033	. 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 3430  {csn 4568  ∪ ciun 4934

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 3063  df-v 3432  df-sn 4569  df-iun 4936

This theorem is referenced by:  iunsuc  6402  funopsn  7093  fparlem3  8055  fparlem4  8056  iunfi  9244  kmlem11  10072  ackbij1lem8  10137  dfid6  14952  fsum2dlem  15694  fsumiun  15745  fprod2dlem  15904  prmreclem4  16848  fiuncmp  23347  ovolfiniun  25446  finiunmbl  25489  volfiniun  25492  voliunlem1  25495  iuninc  32619  cvmliftlem10  35482  mrsubvrs  35710  dfrcl4  44106  iunrelexp0  44132  corclrcl  44137  cotrcltrcl  44155  trclfvdecomr  44158  dfrtrcl4  44168  corcltrcl  44169  cotrclrcl  44172  imaf1hom  49541