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Theorem smoiun 6066
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 3734 . . 3  |-  ( y  e.  U_ x  e.  A  ( B `  x )  <->  E. x  e.  A  y  e.  ( B `  x ) )
2 smofvon 6064 . . . . 5  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( B `  A
)  e.  On )
3 smoel 6065 . . . . . 6  |-  ( ( Smo  B  /\  A  e.  dom  B  /\  x  e.  A )  ->  ( B `  x )  e.  ( B `  A
) )
433expia 1145 . . . . 5  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( x  e.  A  ->  ( B `  x
)  e.  ( B `
 A ) ) )
5 ontr1 4216 . . . . . 6  |-  ( ( B `  A )  e.  On  ->  (
( y  e.  ( B `  x )  /\  ( B `  x )  e.  ( B `  A ) )  ->  y  e.  ( B `  A ) ) )
65expcomd 1375 . . . . 5  |-  ( ( B `  A )  e.  On  ->  (
( B `  x
)  e.  ( B `
 A )  -> 
( y  e.  ( B `  x )  ->  y  e.  ( B `  A ) ) ) )
72, 4, 6sylsyld 57 . . . 4  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( x  e.  A  ->  ( y  e.  ( B `  x )  ->  y  e.  ( B `  A ) ) ) )
87rexlimdv 2488 . . 3  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( E. x  e.  A  y  e.  ( B `  x )  ->  y  e.  ( B `  A ) ) )
91, 8syl5bi 150 . 2  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( y  e.  U_ x  e.  A  ( B `  x )  ->  y  e.  ( B `
 A ) ) )
109ssrdv 3031 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 102    e. wcel 1438   E.wrex 2360    C_ wss 2999   U_ciun 3730   Oncon0 4190   dom cdm 4438   ` cfv 5015   Smo wsmo 6050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-smo 6051
This theorem is referenced by: (None)
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