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Theorem smoiun 6394
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 3925 . . 3  |-  ( y  e.  U_ x  e.  A  ( B `  x )  <->  E. x  e.  A  y  e.  ( B `  x ) )
2 smofvon 6392 . . . . 5  |-  ( ( Smo  B  /\  A  e.  dom  B )  -> 
( B `  A
)  e.  On )
3 smoel 6393 . . . . . 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 4454 . . . . . 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 2679 . . 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 3198 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 1696   E.wrex 2557    C_ wss 3165   U_ciun 3921   Oncon0 4408   dom cdm 4705   ` cfv 5271   Smo wsmo 6378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-tr 4130  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-smo 6379
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