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Theorem iunon 6350
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 4930 . . 3  |-  ( A. x  e.  A  B  e.  On  ->  U_ x  e.  A  B  =  U. ran  (  x  e.  A  |->  B ) )
21adantl 454 . 2  |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  On )  ->  U_ x  e.  A  B  =  U. ran  (  x  e.  A  |->  B ) )
3 mptexg 5706 . . . 4  |-  ( A  e.  V  ->  (
x  e.  A  |->  B )  e.  _V )
4 rnexg 4939 . . . 4  |-  ( ( x  e.  A  |->  B )  e.  _V  ->  ran  (  x  e.  A  |->  B )  e.  _V )
53, 4syl 17 . . 3  |-  ( A  e.  V  ->  ran  (  x  e.  A  |->  B )  e.  _V )
6 eqid 2284 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
76fmpt 5642 . . . 4  |-  ( A. x  e.  A  B  e.  On  <->  ( x  e.  A  |->  B ) : A --> On )
8 frn 5360 . . . 4  |-  ( ( x  e.  A  |->  B ) : A --> On  ->  ran  (  x  e.  A  |->  B )  C_  On )
97, 8sylbi 189 . . 3  |-  ( A. x  e.  A  B  e.  On  ->  ran  (  x  e.  A  |->  B ) 
C_  On )
10 ssonuni 4577 . . . 4  |-  ( ran  (  x  e.  A  |->  B )  e.  _V  ->  ( ran  (  x  e.  A  |->  B ) 
C_  On  ->  U. ran  (  x  e.  A  |->  B )  e.  On ) )
1110imp 420 . . 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 465 . 2  |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  On )  ->  U. ran  (  x  e.  A  |->  B )  e.  On )
132, 12eqeltrd 2358 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 6    /\ wa 360    = wceq 1624    e. wcel 1685   A.wral 2544   _Vcvv 2789    C_ wss 3153   U.cuni 3828   U_ciun 3906    e. cmpt 4078   Oncon0 4391   ran crn 4689   -->wf 5217
This theorem is referenced by:  iunonOLD  6351  oacl  6529  omcl  6530  oecl  6531  rankuni2b  7520  rankval4  7534
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229
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