Theorem iunxsn 4325

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 e. _V
iunxsn.2	|- ( x = A -> B = C )

Assertion

Ref	Expression
iunxsn	|- U_ x e. { 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 e. _V
2		iunxsn.2	. . 3 |- ( x = A -> B = C )
3	2	iunxsng 4324	. 2 |- ( A e. _V -> U_ x e. { A } B = C )
4	1, 3	ax-mp 5	1 |- U_ x e. { A } B = C

Syntax hints:   -> wi 4   = wceq 1437   e. wcel 1872   _Vcvv 3022   {csn 3941   U_ciun 4242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408

This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rex 2720  df-v 3024  df-sbc 3243  df-sn 3942  df-iun 4244

This theorem is referenced by:  iunsuc  5467  fparlem3  6853  fparlem4  6854  iunfi  7815  kmlem11  8541  ackbij1lem8  8608  dfid6  13035  fsum2dlem  13774  fsumiun  13824  fprod2dlem  13977  prmreclem4  14806  fiuncmp  20361  ovolfiniun  22396  finiunmbl  22439  volfiniun  22442  voliunlem1  22445  iuninc  28122  cvmliftlem10  29969  mrsubvrs  30112  dfrcl4  36181  iunrelexp0  36207  corclrcl  36212  cotrcltrcl  36230  trclfvdecomr  36233  dfrtrcl4  36243  corcltrcl  36244  cotrclrcl  36247  funopsn  38824