Theorem iunxsn 5050

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 5049	. 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 2109  Vcvv 3444  {csn 4585  ∪ ciun 4951

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-v 3446  df-sn 4586  df-iun 4953

This theorem is referenced by:  iunsuc  6408  funopsn  7103  fparlem3  8071  fparlem4  8072  iunfi  9271  kmlem11  10093  ackbij1lem8  10158  dfid6  14972  fsum2dlem  15714  fsumiun  15765  fprod2dlem  15924  prmreclem4  16868  fiuncmp  23326  ovolfiniun  25437  finiunmbl  25480  volfiniun  25483  voliunlem1  25486  iuninc  32541  cvmliftlem10  35276  mrsubvrs  35504  dfrcl4  43660  iunrelexp0  43686  corclrcl  43691  cotrcltrcl  43709  trclfvdecomr  43712  dfrtrcl4  43722  corcltrcl  43723  cotrclrcl  43726  imaf1hom  49092