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Theorem iunon 6397
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 4968 . . 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 5786 . . . 4  |-  ( A  e.  V  ->  (
x  e.  A  |->  B )  e.  _V )
4 rnexg 4977 . . . 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 2316 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
76fmpt 5719 . . . 4  |-  ( A. x  e.  A  B  e.  On  <->  ( x  e.  A  |->  B ) : A --> On )
8 frn 5433 . . . 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 4615 . . . 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 2390 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 1633    e. wcel 1701   A.wral 2577   _Vcvv 2822    C_ wss 3186   U.cuni 3864   U_ciun 3942    e. cmpt 4114   Oncon0 4429   ran crn 4727   -->wf 5288
This theorem is referenced by:  iunonOLD  6398  oacl  6576  omcl  6577  oecl  6578  rankuni2b  7570  rankval4  7584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300
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