Theorem iunxsn 5020

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 5019	. 2 ⊢ (𝐴 ∈ V → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶)
4	1, 3	ax-mp 5	1 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {csn 4561  ∪ ciun 4924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-sn 4562  df-iun 4926

This theorem is referenced by:  iunsuc  6348  funopsn  7020  fparlem3  7954  fparlem4  7955  iunfi  9107  kmlem11  9916  ackbij1lem8  9983  dfid6  14739  fsum2dlem  15482  fsumiun  15533  fprod2dlem  15690  prmreclem4  16620  fiuncmp  22555  ovolfiniun  24665  finiunmbl  24708  volfiniun  24711  voliunlem1  24714  iuninc  30900  cvmliftlem10  33256  mrsubvrs  33484  dfrcl4  41284  iunrelexp0  41310  corclrcl  41315  cotrcltrcl  41333  trclfvdecomr  41336  dfrtrcl4  41346  corcltrcl  41347  cotrclrcl  41350