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Theorem iunxsn 3981
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 3980 . 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 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   U_ciun 3905
This theorem is referenced by:  iunsuc  4474  fparlem3  6220  fparlem4  6221  iunfi  7144  kmlem11  7786  ackbij1lem8  7853  fsum2dlem  12233  fsumiun  12279  prmreclem4  12966  fiuncmp  17131  ovolfiniun  18860  finiunmbl  18901  volfiniun  18904  voliunlem1  18907  iuninc  23158  cvmliftlem10  23825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-sn 3646  df-iun 3907
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