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 1570   ∈ wcel 2145  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 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  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  6439  funopsn  7139  funopsnOLD  7140  fparlem3  8108  fparlem4  8109  iunfi  9310  kmlem11  10210  ackbij1lem8  10275  dfid6  15148  fsum2dlem  15903  fsumiun  15955  fprod2dlem  16114  prmreclem4  17058  fiuncmp  23683  ovolfiniun  25783  finiunmbl  25826  volfiniun  25829  voliunlem1  25832  iuninc  33088  cvmliftlem10  35980  mrsubvrs  36208  dfrcl4  44620  iunrelexp0  44646  corclrcl  44651  cotrcltrcl  44669  trclfvdecomr  44672  dfrtrcl4  44682  corcltrcl  44683  cotrclrcl  44686  imaf1hom  50138