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Theorem iunxsng 4414
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 4337 . . 3  |-  ( y  e.  U_ x  e. 
{ A } B  <->  E. x  e.  { A } y  e.  B
)
2 iunxsng.1 . . . . 5  |-  ( x  =  A  ->  B  =  C )
32eleq2d 2527 . . . 4  |-  ( x  =  A  ->  (
y  e.  B  <->  y  e.  C ) )
43rexsng 4068 . . 3  |-  ( A  e.  V  ->  ( E. x  e.  { A } y  e.  B  <->  y  e.  C ) )
51, 4syl5bb 257 . 2  |-  ( A  e.  V  ->  (
y  e.  U_ x  e.  { A } B  <->  y  e.  C ) )
65eqrdv 2454 1  |-  ( A  e.  V  ->  U_ x  e.  { A } B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   E.wrex 2808   {csn 4032   U_ciun 4332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328  df-sn 4033  df-iun 4334
This theorem is referenced by:  iunxsn  4415  iunxprg  32566
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