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  7096  fparlem3  8059  fparlem4  8060  iunfi  9248  kmlem11  10076  ackbij1lem8  10141  dfid6  14956  fsum2dlem  15698  fsumiun  15749  fprod2dlem  15908  prmreclem4  16852  fiuncmp  23353  ovolfiniun  25463  finiunmbl  25506  volfiniun  25509  voliunlem1  25512  iuninc  32639  cvmliftlem10  35501  mrsubvrs  35729  dfrcl4  43995  iunrelexp0  44021  corclrcl  44026  cotrcltrcl  44044  trclfvdecomr  44047  dfrtrcl4  44057  corcltrcl  44058  cotrclrcl  44061  imaf1hom  49430