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Theorem smofvon2 6547
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 6538 . . . 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 972 . . 3  |-  ( Smo 
F  ->  F : dom  F --> On )
3 ffvelrn 5800 . . . 4  |-  ( ( F : dom  F --> On  /\  B  e.  dom  F )  ->  ( F `  B )  e.  On )
43expcom 425 . . 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 5688 . . . 4  |-  ( -.  B  e.  dom  F  ->  ( F `  B
)  =  (/) )
7 0elon 4568 . . . 4  |-  (/)  e.  On
86, 7syl6eqel 2468 . . 3  |-  ( -.  B  e.  dom  F  ->  ( F `  B
)  e.  On )
98a1d 23 . 2  |-  ( -.  B  e.  dom  F  ->  ( Smo  F  -> 
( F `  B
)  e.  On ) )
105, 9pm2.61i 158 1  |-  ( Smo 
F  ->  ( F `  B )  e.  On )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1717   A.wral 2642   (/)c0 3564   Ord word 4514   Oncon0 4515   dom cdm 4811   -->wf 5383   ` cfv 5387   Smo wsmo 6536
This theorem is referenced by:  smo11  6555  smoord  6556  smoword  6557  smogt  6558
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-tr 4237  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-smo 6537
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