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Theorem smofvon2dm 6322
Description: The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
smofvon2dm  |-  ( ( Smo  F  /\  B  e.  dom  F )  -> 
( F `  B
)  e.  On )

Proof of Theorem smofvon2dm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsmo2 6313 . . 3  |-  ( Smo 
F  <->  ( F : dom  F --> On  /\  Ord  dom 
F  /\  A. x  e.  dom  F A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
21simp1bi 1014 . 2  |-  ( Smo 
F  ->  F : dom  F --> On )
32ffvelcdmda 5672 1  |-  ( ( Smo  F  /\  B  e.  dom  F )  -> 
( F `  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2160   A.wral 2468   Ord word 4380   Oncon0 4381   dom cdm 4644   -->wf 5231   ` cfv 5235   Smo wsmo 6311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-tr 4117  df-id 4311  df-iord 4384  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-smo 6312
This theorem is referenced by: (None)
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