MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfiun3g Structured version   Unicode version

Theorem dfiun3g 5049
Description: Alternate definition of indexed union when  B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
dfiun3g  |-  ( A. x  e.  A  B  e.  C  ->  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )

Proof of Theorem dfiun3g
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfiun2g 4274 . 2  |-  ( A. x  e.  A  B  e.  C  ->  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B } )
2 eqid 2428 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
32rnmpt 5042 . . 3  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
43unieqi 4171 . 2  |-  U. ran  ( x  e.  A  |->  B )  =  U. { y  |  E. x  e.  A  y  =  B }
51, 4syl6eqr 2480 1  |-  ( A. x  e.  A  B  e.  C  ->  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872   {cab 2414   A.wral 2714   E.wrex 2715   U.cuni 4162   U_ciun 4242    |-> cmpt 4425   ran crn 4797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-cnv 4804  df-dm 4806  df-rn 4807
This theorem is referenced by:  dfiun3  5051  iunon  7012  onoviun  7017  gruiun  9175  tgiun  19937  acunirnmpt2f  28209  locfinreflem  28619  carsgclctunlem2  29103  pmeasadd  29110  saliuncl  38047  meadjiun  38155  omeiunle  38189
  Copyright terms: Public domain W3C validator