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Theorem rniun 4677
Description: The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
rniun  ran  U_  U_  ran

Proof of Theorem rniun
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 2571 . . . 4  <. ,  >.  <. ,  >.
2 vex 2554 . . . . . 6 
_V
32elrn2 4519 . . . . 5  ran  <. ,  >.
43rexbii 2325 . . . 4  ran  <. ,  >.
5 eliun 3652 . . . . 5  <. ,  >.  U_  <. ,  >.
65exbii 1493 . . . 4  <. , 
>.  U_  <. ,  >.
71, 4, 63bitr4ri 202 . . 3  <. , 
>.  U_  ran
82elrn2 4519 . . 3  ran  U_  <. ,  >. 
U_
9 eliun 3652 . . 3  U_  ran  ran
107, 8, 93bitr4i 201 . 2  ran  U_ 
U_  ran
1110eqriv 2034 1  ran  U_  U_  ran
Colors of variables: wff set class
Syntax hints:   wceq 1242  wex 1378   wcel 1390  wrex 2301   <.cop 3370   U_ciun 3648   ran crn 4289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  rnuni  4678  fun11iun  5090
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