Theorem iunxsn 5056

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 5055	. 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 2142  Vcvv 3454  {csn 4588  ∪ ciun 4955

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-v 3456  df-sn 4589  df-iun 4957

This theorem is used by:  iunsuc  6448  funopsn  7144  funopsnOLD  7145  fparlem3  8107  fparlem4  8108  iunfi  9298  kmlem11  10151  ackbij1lem8  10216  dfid6  15072  fsum2dlem  15828  fsumiun  15880  fprod2dlem  16041  prmreclem4  16985  fiuncmp  23572  ovolfiniun  25671  finiunmbl  25714  volfiniun  25717  voliunlem1  25720  iuninc  32916  cvmliftlem10  35794  mrsubvrs  36022  dfrcl4  44430  iunrelexp0  44456  corclrcl  44461  cotrcltrcl  44479  trclfvdecomr  44482  dfrtrcl4  44492  corcltrcl  44493  cotrclrcl  44496  imaf1hom  49914