MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfiun3 Structured version   Unicode version

Theorem dfiun3 5126
Description: Alternate definition of indexed union when  B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfiun3.1  |-  B  e. 
_V
Assertion
Ref Expression
dfiun3  |-  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B )

Proof of Theorem dfiun3
StepHypRef Expression
1 dfiun3g 5124 . 2  |-  ( A. x  e.  A  B  e.  _V  ->  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )
2 dfiun3.1 . . 3  |-  B  e. 
_V
32a1i 11 . 2  |-  ( x  e.  A  ->  B  e.  _V )
41, 3mprg 2777 1  |-  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   _Vcvv 2958   U.cuni 4017   U_ciun 4095    e. cmpt 4268   ran crn 4881
This theorem is referenced by:  tgrest  17225  dstfrvunirn  24734  mblfinlem2  26246  volsupnfl  26253  comppfsc  26389  istotbnd3  26482  sstotbnd  26486
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-cnv 4888  df-dm 4890  df-rn 4891
  Copyright terms: Public domain W3C validator