Theorem iunxsn 5096

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 5095	. 2 ⊢ (𝐴 ∈ V → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶)
4	1, 3	ax-mp 5	1 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  Vcvv 3478  {csn 4631  ∪ ciun 4996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-sn 4632  df-iun 4998

This theorem is referenced by:  iunsuc  6471  funopsn  7168  fparlem3  8138  fparlem4  8139  iunfi  9381  kmlem11  10199  ackbij1lem8  10264  dfid6  15064  fsum2dlem  15803  fsumiun  15854  fprod2dlem  16013  prmreclem4  16953  fiuncmp  23428  ovolfiniun  25550  finiunmbl  25593  volfiniun  25596  voliunlem1  25599  iuninc  32581  cvmliftlem10  35279  mrsubvrs  35507  dfrcl4  43666  iunrelexp0  43692  corclrcl  43697  cotrcltrcl  43715  trclfvdecomr  43718  dfrtrcl4  43728  corcltrcl  43729  cotrclrcl  43732