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

Theorem iunon 6355
Description: The indexed union of a set of ordinal numbers  B ( x ) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.)
Assertion
Ref Expression
iunon  |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  On )  ->  U_ x  e.  A  B  e.  On )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem iunon
StepHypRef Expression
1 dfiun3g 4931 . . 3  |-  ( A. x  e.  A  B  e.  On  ->  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )
21adantl 452 . 2  |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  On )  ->  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )
3 mptexg 5745 . . . 4  |-  ( A  e.  V  ->  (
x  e.  A  |->  B )  e.  _V )
4 rnexg 4940 . . . 4  |-  ( ( x  e.  A  |->  B )  e.  _V  ->  ran  ( x  e.  A  |->  B )  e.  _V )
53, 4syl 15 . . 3  |-  ( A  e.  V  ->  ran  ( x  e.  A  |->  B )  e.  _V )
6 eqid 2283 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
76fmpt 5681 . . . 4  |-  ( A. x  e.  A  B  e.  On  <->  ( x  e.  A  |->  B ) : A --> On )
8 frn 5395 . . . 4  |-  ( ( x  e.  A  |->  B ) : A --> On  ->  ran  ( x  e.  A  |->  B )  C_  On )
97, 8sylbi 187 . . 3  |-  ( A. x  e.  A  B  e.  On  ->  ran  ( x  e.  A  |->  B ) 
C_  On )
10 ssonuni 4578 . . . 4  |-  ( ran  ( x  e.  A  |->  B )  e.  _V  ->  ( ran  ( x  e.  A  |->  B ) 
C_  On  ->  U. ran  ( x  e.  A  |->  B )  e.  On ) )
1110imp 418 . . 3  |-  ( ( ran  ( x  e.  A  |->  B )  e. 
_V  /\  ran  ( x  e.  A  |->  B ) 
C_  On )  ->  U. ran  ( x  e.  A  |->  B )  e.  On )
125, 9, 11syl2an 463 . 2  |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  On )  ->  U. ran  ( x  e.  A  |->  B )  e.  On )
132, 12eqeltrd 2357 1  |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  On )  ->  U_ x  e.  A  B  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   U.cuni 3827   U_ciun 3905    e. cmpt 4077   Oncon0 4392   ran crn 4690   -->wf 5251
This theorem is referenced by:  iunonOLD  6356  oacl  6534  omcl  6535  oecl  6536  rankuni2b  7525  rankval4  7539
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
  Copyright terms: Public domain W3C validator