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Theorem iunxsn 3997
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
StepHypRef Expression
1 iunxsn.1 . 2  |-  A  e. 
_V
2 iunxsn.2 . . 3  |-  ( x  =  A  ->  B  =  C )
32iunxsng 3996 . 2  |-  ( A  e.  _V  ->  U_ x  e.  { A } B  =  C )
41, 3ax-mp 8 1  |-  U_ x  e.  { A } B  =  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   U_ciun 3921
This theorem is referenced by:  iunsuc  4490  fparlem3  6236  fparlem4  6237  iunfi  7160  kmlem11  7802  ackbij1lem8  7869  fsum2dlem  12249  fsumiun  12295  prmreclem4  12982  fiuncmp  17147  ovolfiniun  18876  finiunmbl  18917  volfiniun  18920  voliunlem1  18923  iuninc  23174  cvmliftlem10  23840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005  df-sn 3659  df-iun 3923
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