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Theorem smoiun 5916
Description: The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.)
Assertion
Ref Expression
smoiun  |-  ( ( Smo  B  /\  A  e.  dom  B )  ->  U_ x  e.  A  ( B `  x ) 
C_  ( B `  A ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem smoiun
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eliun 3661 . . 3  |-  ( y  e.  U_ x  e.  A  ( B `  x )  <->  E. x  e.  A  y  e.  ( B `  x ) )
2 smofvon 5914 . . . . 5  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( B `  A
)  e.  On )
3 smoel 5915 . . . . . 6  |-  ( ( Smo  B  /\  A  e.  dom  B  /\  x  e.  A )  ->  ( B `  x )  e.  ( B `  A
) )
433expia 1106 . . . . 5  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( x  e.  A  ->  ( B `  x
)  e.  ( B `
 A ) ) )
5 ontr1 4126 . . . . . 6  |-  ( ( B `  A )  e.  On  ->  (
( y  e.  ( B `  x )  /\  ( B `  x )  e.  ( B `  A ) )  ->  y  e.  ( B `  A ) ) )
65expcomd 1330 . . . . 5  |-  ( ( B `  A )  e.  On  ->  (
( B `  x
)  e.  ( B `
 A )  -> 
( y  e.  ( B `  x )  ->  y  e.  ( B `  A ) ) ) )
72, 4, 6sylsyld 52 . . . 4  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( x  e.  A  ->  ( y  e.  ( B `  x )  ->  y  e.  ( B `  A ) ) ) )
87rexlimdv 2432 . . 3  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( E. x  e.  A  y  e.  ( B `  x )  ->  y  e.  ( B `  A ) ) )
91, 8syl5bi 141 . 2  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( y  e.  U_ x  e.  A  ( B `  x )  ->  y  e.  ( B `
 A ) ) )
109ssrdv 2951 1  |-  ( ( Smo  B  /\  A  e.  dom  B )  ->  U_ x  e.  A  ( B `  x ) 
C_  ( B `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    e. wcel 1393   E.wrex 2307    C_ wss 2917   U_ciun 3657   Oncon0 4100   dom cdm 4345   ` cfv 4902   Smo wsmo 5900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-smo 5901
This theorem is referenced by: (None)
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