Theorem iunxsn 5091

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 5090	. 2 ⊢ (𝐴 ∈ V → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶)
4	1, 3	ax-mp 5	1 ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3480  {csn 4626  ∪ ciun 4991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-sn 4627  df-iun 4993

This theorem is referenced by:  iunsuc  6469  funopsn  7168  fparlem3  8139  fparlem4  8140  iunfi  9383  kmlem11  10201  ackbij1lem8  10266  dfid6  15067  fsum2dlem  15806  fsumiun  15857  fprod2dlem  16016  prmreclem4  16957  fiuncmp  23412  ovolfiniun  25536  finiunmbl  25579  volfiniun  25582  voliunlem1  25585  iuninc  32573  cvmliftlem10  35299  mrsubvrs  35527  dfrcl4  43689  iunrelexp0  43715  corclrcl  43720  cotrcltrcl  43738  trclfvdecomr  43741  dfrtrcl4  43751  corcltrcl  43752  cotrclrcl  43755