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Theorem smoiun 6378
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 3909 . . 3  |-  ( y  e.  U_ x  e.  A  ( B `  x )  <->  E. x  e.  A  y  e.  ( B `  x ) )
2 smofvon 6376 . . . . 5  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( B `  A
)  e.  On )
3 smoel 6377 . . . . . 6  |-  ( ( Smo  B  /\  A  e.  dom  B  /\  x  e.  A )  ->  ( B `  x )  e.  ( B `  A
) )
433expia 1153 . . . . 5  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( x  e.  A  ->  ( B `  x
)  e.  ( B `
 A ) ) )
5 ontr1 4438 . . . . . 6  |-  ( ( B `  A )  e.  On  ->  (
( y  e.  ( B `  x )  /\  ( B `  x )  e.  ( B `  A ) )  ->  y  e.  ( B `  A ) ) )
65exp3acom23 1362 . . . . 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 2666 . . 3  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( E. x  e.  A  y  e.  ( B `  x )  ->  y  e.  ( B `  A ) ) )
91, 8syl5bi 208 . 2  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( y  e.  U_ x  e.  A  ( B `  x )  ->  y  e.  ( B `
 A ) ) )
109ssrdv 3185 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 358    e. wcel 1684   E.wrex 2544    C_ wss 3152   U_ciun 3905   Oncon0 4392   dom cdm 4689   ` cfv 5255   Smo wsmo 6362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-tr 4114  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-smo 6363
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