Theorem iunxsn 4046

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 4045	. 2 ⊢ (A ∈ V → ∪x ∈ {A}B = C)
4	1, 3	ax-mp 5	1 ⊢ ∪x ∈ {A}B = C

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 2860  {csn 3738  ∪ciun 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-sn 3742  df-iun 3972

This theorem is referenced by: (None)