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Theorem smofvon2 6930
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 6921 . . . 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 1003 . . 3  |-  ( Smo 
F  ->  F : dom  F --> On )
3 ffvelrn 5953 . . . 4  |-  ( ( F : dom  F --> On  /\  B  e.  dom  F )  ->  ( F `  B )  e.  On )
43expcom 435 . . 3  |-  ( B  e.  dom  F  -> 
( F : dom  F --> On  ->  ( F `  B )  e.  On ) )
52, 4syl5 32 . 2  |-  ( B  e.  dom  F  -> 
( Smo  F  ->  ( F `  B )  e.  On ) )
6 ndmfv 5826 . . . 4  |-  ( -.  B  e.  dom  F  ->  ( F `  B
)  =  (/) )
7 0elon 4883 . . . 4  |-  (/)  e.  On
86, 7syl6eqel 2550 . . 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 1758   A.wral 2799   (/)c0 3748   Ord word 4829   Oncon0 4830   dom cdm 4951   -->wf 5525   ` cfv 5529   Smo wsmo 6919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-tr 4497  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-smo 6920
This theorem is referenced by:  smo11  6938  smoord  6939  smoword  6940  smogt  6941
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