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

Theorem iunxsn 4405
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.)
Hypotheses
Ref Expression
iunxsn.1  |-  A  e. 
_V
iunxsn.2  |-  ( x  =  A  ->  B  =  C )
Assertion
Ref Expression
iunxsn  |-  U_ x  e.  { A } B  =  C
Distinct variable groups:    x, A    x, C
Allowed substitution hint:    B( x)

Proof of Theorem iunxsn
StepHypRef Expression
1 iunxsn.1 . 2  |-  A  e. 
_V
2 iunxsn.2 . . 3  |-  ( x  =  A  ->  B  =  C )
32iunxsng 4404 . 2  |-  ( A  e.  _V  ->  U_ x  e.  { A } B  =  C )
41, 3ax-mp 5 1  |-  U_ x  e.  { A } B  =  C
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027   U_ciun 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115  df-sbc 3332  df-sn 4028  df-iun 4327
This theorem is referenced by:  iunsuc  4960  fparlem3  6882  fparlem4  6883  iunfi  7804  kmlem11  8536  ackbij1lem8  8603  fsum2dlem  13544  fsumiun  13594  prmreclem4  14292  fiuncmp  19670  ovolfiniun  21647  finiunmbl  21689  volfiniun  21692  voliunlem1  21695  iuninc  27101  cvmliftlem10  28379  fprod2dlem  28687
  Copyright terms: Public domain W3C validator