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Theorem iunxsng 4162
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 4090 . . 3  |-  ( y  e.  U_ x  e. 
{ A } B  <->  E. x  e.  { A } y  e.  B
)
2 iunxsng.1 . . . . 5  |-  ( x  =  A  ->  B  =  C )
32eleq2d 2503 . . . 4  |-  ( x  =  A  ->  (
y  e.  B  <->  y  e.  C ) )
43rexsng 3840 . . 3  |-  ( A  e.  V  ->  ( E. x  e.  { A } y  e.  B  <->  y  e.  C ) )
51, 4syl5bb 249 . 2  |-  ( A  e.  V  ->  (
y  e.  U_ x  e.  { A } B  <->  y  e.  C ) )
65eqrdv 2434 1  |-  ( A  e.  V  ->  U_ x  e.  { A } B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   E.wrex 2699   {csn 3807   U_ciun 4086
This theorem is referenced by:  iunxsn  4163  iunxprg  28059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2703  df-rex 2704  df-v 2951  df-sbc 3155  df-sn 3813  df-iun 4088
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