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Theorem smoiun 6625
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 4099 . . 3  |-  ( y  e.  U_ x  e.  A  ( B `  x )  <->  E. x  e.  A  y  e.  ( B `  x ) )
2 smofvon 6623 . . . . 5  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( B `  A
)  e.  On )
3 smoel 6624 . . . . . 6  |-  ( ( Smo  B  /\  A  e.  dom  B  /\  x  e.  A )  ->  ( B `  x )  e.  ( B `  A
) )
433expia 1156 . . . . 5  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( x  e.  A  ->  ( B `  x
)  e.  ( B `
 A ) ) )
5 ontr1 4629 . . . . . 6  |-  ( ( B `  A )  e.  On  ->  (
( y  e.  ( B `  x )  /\  ( B `  x )  e.  ( B `  A ) )  ->  y  e.  ( B `  A ) ) )
65exp3acom23 1382 . . . . 5  |-  ( ( B `  A )  e.  On  ->  (
( B `  x
)  e.  ( B `
 A )  -> 
( y  e.  ( B `  x )  ->  y  e.  ( B `  A ) ) ) )
72, 4, 6sylsyld 55 . . . 4  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( x  e.  A  ->  ( y  e.  ( B `  x )  ->  y  e.  ( B `  A ) ) ) )
87rexlimdv 2831 . . 3  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( E. x  e.  A  y  e.  ( B `  x )  ->  y  e.  ( B `  A ) ) )
91, 8syl5bi 210 . 2  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( y  e.  U_ x  e.  A  ( B `  x )  ->  y  e.  ( B `
 A ) ) )
109ssrdv 3356 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 360    e. wcel 1726   E.wrex 2708    C_ wss 3322   U_ciun 4095   Oncon0 4583   dom cdm 4880   ` cfv 5456   Smo wsmo 6609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-tr 4305  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-smo 6610
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