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Theorem smofvon2 7072
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 7063 . . . 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 1022 . . 3  |-  ( Smo 
F  ->  F : dom  F --> On )
3 ffvelrn 6018 . . . 4  |-  ( ( F : dom  F --> On  /\  B  e.  dom  F )  ->  ( F `  B )  e.  On )
43expcom 437 . . 3  |-  ( B  e.  dom  F  -> 
( F : dom  F --> On  ->  ( F `  B )  e.  On ) )
52, 4syl5 33 . 2  |-  ( B  e.  dom  F  -> 
( Smo  F  ->  ( F `  B )  e.  On ) )
6 ndmfv 5887 . . . 4  |-  ( -.  B  e.  dom  F  ->  ( F `  B
)  =  (/) )
7 0elon 5475 . . . 4  |-  (/)  e.  On
86, 7syl6eqel 2536 . . 3  |-  ( -.  B  e.  dom  F  ->  ( F `  B
)  e.  On )
98a1d 26 . 2  |-  ( -.  B  e.  dom  F  ->  ( Smo  F  -> 
( F `  B
)  e.  On ) )
105, 9pm2.61i 168 1  |-  ( Smo 
F  ->  ( F `  B )  e.  On )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1886   A.wral 2736   (/)c0 3730   dom cdm 4833   Ord word 5421   Oncon0 5422   -->wf 5577   ` cfv 5581   Smo wsmo 7061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-tr 4497  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-ord 5425  df-on 5426  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-smo 7062
This theorem is referenced by:  smo11  7080  smoord  7081  smoword  7082  smogt  7083
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