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Theorem iunxsngf 3890
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.) (Revised by Thierry Arnoux, 2-May-2020.)
Hypotheses
Ref Expression
iunxsngf.1  |-  F/_ x C
iunxsngf.2  |-  ( x  =  A  ->  B  =  C )
Assertion
Ref Expression
iunxsngf  |-  ( A  e.  V  ->  U_ x  e.  { A } B  =  C )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    C( x)    V( x)

Proof of Theorem iunxsngf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eliun 3817 . . 3  |-  ( y  e.  U_ x  e. 
{ A } B  <->  E. x  e.  { A } y  e.  B
)
2 rexsns 3563 . . . 4  |-  ( E. x  e.  { A } y  e.  B  <->  [. A  /  x ]. y  e.  B )
3 iunxsngf.1 . . . . . 6  |-  F/_ x C
43nfcri 2275 . . . . 5  |-  F/ x  y  e.  C
5 iunxsngf.2 . . . . . 6  |-  ( x  =  A  ->  B  =  C )
65eleq2d 2209 . . . . 5  |-  ( x  =  A  ->  (
y  e.  B  <->  y  e.  C ) )
74, 6sbciegf 2940 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. y  e.  B  <->  y  e.  C ) )
82, 7syl5bb 191 . . 3  |-  ( A  e.  V  ->  ( E. x  e.  { A } y  e.  B  <->  y  e.  C ) )
91, 8syl5bb 191 . 2  |-  ( A  e.  V  ->  (
y  e.  U_ x  e.  { A } B  <->  y  e.  C ) )
109eqrdv 2137 1  |-  ( A  e.  V  ->  U_ x  e.  { A } B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   F/_wnfc 2268   E.wrex 2417   [.wsbc 2909   {csn 3527   U_ciun 3813
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-sn 3533  df-iun 3815
This theorem is referenced by:  iunfidisj  6834  iuncld  12298
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