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Theorem iunxsng 3854
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
Hypothesis
Ref Expression
iunxsng.1  |-  ( x  =  A  ->  B  =  C )
Assertion
Ref Expression
iunxsng  |-  ( A  e.  V  ->  U_ x  e.  { A } B  =  C )
Distinct variable groups:    x, A    x, C
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem iunxsng
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eliun 3783 . . 3  |-  ( y  e.  U_ x  e. 
{ A } B  <->  E. x  e.  { A } y  e.  B
)
2 iunxsng.1 . . . . 5  |-  ( x  =  A  ->  B  =  C )
32eleq2d 2184 . . . 4  |-  ( x  =  A  ->  (
y  e.  B  <->  y  e.  C ) )
43rexsng 3531 . . 3  |-  ( A  e.  V  ->  ( E. x  e.  { A } y  e.  B  <->  y  e.  C ) )
51, 4syl5bb 191 . 2  |-  ( A  e.  V  ->  (
y  e.  U_ x  e.  { A } B  <->  y  e.  C ) )
65eqrdv 2113 1  |-  ( A  e.  V  ->  U_ x  e.  { A } B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463   E.wrex 2391   {csn 3493   U_ciun 3779
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 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 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-sn 3499  df-iun 3781
This theorem is referenced by:  iunxsn  3855  iunxprg  3859  rdgisuc1  6235  oasuc  6314  omsuc  6322
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