Theorem iunxsn 5095

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 5094	. 2 ⊢ (𝐴 ∈ V → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶)
4	1, 3	ax-mp 5	1 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3461  {csn 4630  ∪ ciun 4997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-v 3463  df-sn 4631  df-iun 4999

This theorem is referenced by:  iunsuc  6456  funopsn  7157  fparlem3  8119  fparlem4  8120  iunfi  9366  kmlem11  10185  ackbij1lem8  10252  dfid6  15011  fsum2dlem  15752  fsumiun  15803  fprod2dlem  15960  prmreclem4  16891  fiuncmp  23352  ovolfiniun  25474  finiunmbl  25517  volfiniun  25520  voliunlem1  25523  iuninc  32430  cvmliftlem10  35035  mrsubvrs  35263  dfrcl4  43248  iunrelexp0  43274  corclrcl  43279  cotrcltrcl  43297  trclfvdecomr  43300  dfrtrcl4  43310  corcltrcl  43311  cotrclrcl  43314