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Theorem smofvon2 7060
Description: The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
smofvon2  |-  ( Smo 
F  ->  ( F `  B )  e.  On )

Proof of Theorem smofvon2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsmo2 7051 . . . 4  |-  ( Smo 
F  <->  ( F : dom  F --> On  /\  Ord  dom 
F  /\  A. x  e.  dom  F A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
21simp1bi 1012 . . 3  |-  ( Smo 
F  ->  F : dom  F --> On )
3 ffvelrn 6007 . . . 4  |-  ( ( F : dom  F --> On  /\  B  e.  dom  F )  ->  ( F `  B )  e.  On )
43expcom 433 . . 3  |-  ( B  e.  dom  F  -> 
( F : dom  F --> On  ->  ( F `  B )  e.  On ) )
52, 4syl5 30 . 2  |-  ( B  e.  dom  F  -> 
( Smo  F  ->  ( F `  B )  e.  On ) )
6 ndmfv 5873 . . . 4  |-  ( -.  B  e.  dom  F  ->  ( F `  B
)  =  (/) )
7 0elon 5463 . . . 4  |-  (/)  e.  On
86, 7syl6eqel 2498 . . 3  |-  ( -.  B  e.  dom  F  ->  ( F `  B
)  e.  On )
98a1d 25 . 2  |-  ( -.  B  e.  dom  F  ->  ( Smo  F  -> 
( F `  B
)  e.  On ) )
105, 9pm2.61i 164 1  |-  ( Smo 
F  ->  ( F `  B )  e.  On )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1842   A.wral 2754   (/)c0 3738   dom cdm 4823   Ord word 5409   Oncon0 5410   -->wf 5565   ` cfv 5569   Smo wsmo 7049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-tr 4490  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-ord 5413  df-on 5414  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577  df-smo 7050
This theorem is referenced by:  smo11  7068  smoord  7069  smoword  7070  smogt  7071
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