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Theorem smofvon2 7024
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 7015 . . . 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 1011 . . 3  |-  ( Smo 
F  ->  F : dom  F --> On )
3 ffvelrn 6017 . . . 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 5888 . . . 4  |-  ( -.  B  e.  dom  F  ->  ( F `  B
)  =  (/) )
7 0elon 4931 . . . 4  |-  (/)  e.  On
86, 7syl6eqel 2563 . . 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 1767   A.wral 2814   (/)c0 3785   Ord word 4877   Oncon0 4878   dom cdm 4999   -->wf 5582   ` cfv 5586   Smo wsmo 7013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-smo 7014
This theorem is referenced by:  smo11  7032  smoord  7033  smoword  7034  smogt  7035
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