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Theorem iunxsng 3980
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 3909 . . 3  |-  ( y  e.  U_ x  e. 
{ A } B  <->  E. x  e.  { A } y  e.  B
)
2 iunxsng.1 . . . . 5  |-  ( x  =  A  ->  B  =  C )
32eleq2d 2350 . . . 4  |-  ( x  =  A  ->  (
y  e.  B  <->  y  e.  C ) )
43rexsng 3673 . . 3  |-  ( A  e.  V  ->  ( E. x  e.  { A } y  e.  B  <->  y  e.  C ) )
51, 4syl5bb 248 . 2  |-  ( A  e.  V  ->  (
y  e.  U_ x  e.  { A } B  <->  y  e.  C ) )
65eqrdv 2281 1  |-  ( A  e.  V  ->  U_ x  e.  { A } B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   E.wrex 2544   {csn 3640   U_ciun 3905
This theorem is referenced by:  iunxsn  3981
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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