Theorem iunxsn 5058

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 5057	. 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 3450  {csn 4592  ∪ ciun 4958

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 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-v 3452  df-sn 4593  df-iun 4960

This theorem is referenced by:  iunsuc  6422  funopsn  7123  fparlem3  8096  fparlem4  8097  iunfi  9301  kmlem11  10121  ackbij1lem8  10186  dfid6  15001  fsum2dlem  15743  fsumiun  15794  fprod2dlem  15953  prmreclem4  16897  fiuncmp  23298  ovolfiniun  25409  finiunmbl  25452  volfiniun  25455  voliunlem1  25458  iuninc  32496  cvmliftlem10  35288  mrsubvrs  35516  dfrcl4  43672  iunrelexp0  43698  corclrcl  43703  cotrcltrcl  43721  trclfvdecomr  43724  dfrtrcl4  43734  corcltrcl  43735  cotrclrcl  43738  imaf1hom  49101