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

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  {csn 4583  ∪ ciun 4950

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-v 3452  df-sn 4584  df-iun 4952

This theorem is used by:  iunsuc  6450  funopsn  7151  funopsnOLD  7152  fparlem3  8125  fparlem4  8126  iunfi  9332  kmlem11  10239  ackbij1lem8  10304  dfid6  15181  fsum2dlem  15936  fsumiun  15988  fprod2dlem  16147  prmreclem4  17097  fiuncmp  23722  ovolfiniun  25822  finiunmbl  25865  volfiniun  25868  voliunlem1  25871  iuninc  33197  cvmliftlem10  36101  mrsubvrs  36329  dfrcl4  44719  iunrelexp0  44745  corclrcl  44750  cotrcltrcl  44768  trclfvdecomr  44771  dfrtrcl4  44781  corcltrcl  44782  cotrclrcl  44785  imaf1hom  50259