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Theorem iunxsn 3857
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 3856 . 2  |-  ( A  e.  _V  ->  U_ x  e.  { A } B  =  C )
41, 3ax-mp 5 1  |-  U_ x  e.  { A } B  =  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463   _Vcvv 2658   {csn 3495   U_ciun 3781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-sn 3501  df-iun 3783
This theorem is referenced by:  iunsuc  4310  fsum2dlemstep  11143  fsumiun  11186
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