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Theorem iunon 7001
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 5248 . . 3  |-  ( A. x  e.  A  B  e.  On  ->  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )
21adantl 466 . 2  |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  On )  ->  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )
3 mptexg 6123 . . . 4  |-  ( A  e.  V  ->  (
x  e.  A  |->  B )  e.  _V )
4 rnexg 6708 . . . 4  |-  ( ( x  e.  A  |->  B )  e.  _V  ->  ran  ( x  e.  A  |->  B )  e.  _V )
53, 4syl 16 . . 3  |-  ( A  e.  V  ->  ran  ( x  e.  A  |->  B )  e.  _V )
6 eqid 2462 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
76fmpt 6035 . . . 4  |-  ( A. x  e.  A  B  e.  On  <->  ( x  e.  A  |->  B ) : A --> On )
8 frn 5730 . . . 4  |-  ( ( x  e.  A  |->  B ) : A --> On  ->  ran  ( x  e.  A  |->  B )  C_  On )
97, 8sylbi 195 . . 3  |-  ( A. x  e.  A  B  e.  On  ->  ran  ( x  e.  A  |->  B ) 
C_  On )
10 ssonuni 6595 . . . 4  |-  ( ran  ( x  e.  A  |->  B )  e.  _V  ->  ( ran  ( x  e.  A  |->  B ) 
C_  On  ->  U. ran  ( x  e.  A  |->  B )  e.  On ) )
1110imp 429 . . 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 477 . 2  |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  On )  ->  U. ran  ( x  e.  A  |->  B )  e.  On )
132, 12eqeltrd 2550 1  |-  ( ( A  e.  V  /\  A. x  e.  A  B  e.  On )  ->  U_ x  e.  A  B  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809   _Vcvv 3108    C_ wss 3471   U.cuni 4240   U_ciun 4320    |-> cmpt 4500   Oncon0 4873   ran crn 4995   -->wf 5577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589
This theorem is referenced by:  iunonOLD  7002  oacl  7177  omcl  7178  oecl  7179  rankuni2b  8262  rankval4  8276
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