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Theorem iunon 6323
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 4919 . . 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 5679 . . . 4  |-  ( A  e.  V  ->  (
x  e.  A  |->  B )  e.  _V )
4 rnexg 4928 . . . 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 2258 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
76fmpt 5615 . . . 4  |-  ( A. x  e.  A  B  e.  On  <->  ( x  e.  A  |->  B ) : A --> On )
8 frn 5333 . . . 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 4550 . . . 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 2332 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 1619    e. wcel 1621   A.wral 2518   _Vcvv 2763    C_ wss 3127   U.cuni 3801   U_ciun 3879    e. cmpt 4051   Oncon0 4364   ran crn 4662   -->wf 4669
This theorem is referenced by:  iunonOLD  6324  oacl  6502  omcl  6503  oecl  6504  rankuni2b  7493  rankval4  7507
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689
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