MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunxsng Structured version   Visualization version   Unicode version

Theorem iunxsng 4359
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 4282 . . 3  |-  ( y  e.  U_ x  e. 
{ A } B  <->  E. x  e.  { A } y  e.  B
)
2 iunxsng.1 . . . . 5  |-  ( x  =  A  ->  B  =  C )
32eleq2d 2513 . . . 4  |-  ( x  =  A  ->  (
y  e.  B  <->  y  e.  C ) )
43rexsng 4006 . . 3  |-  ( A  e.  V  ->  ( E. x  e.  { A } y  e.  B  <->  y  e.  C ) )
51, 4syl5bb 261 . 2  |-  ( A  e.  V  ->  (
y  e.  U_ x  e.  { A } B  <->  y  e.  C ) )
65eqrdv 2448 1  |-  ( A  e.  V  ->  U_ x  e.  { A } B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1443    e. wcel 1886   E.wrex 2737   {csn 3967   U_ciun 4277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ral 2741  df-rex 2742  df-v 3046  df-sbc 3267  df-sn 3968  df-iun 4279
This theorem is referenced by:  iunxsn  4360  disjiun2  37392  carageniuncllem1  38336  caratheodorylem1  38341  iunxprg  38998
  Copyright terms: Public domain W3C validator