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Theorem smofvon2dm 5945
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 5936 . . 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 954 . 2  |-  ( Smo 
F  ->  F : dom  F --> On )
32ffvelrnda 5334 1  |-  ( ( Smo  F  /\  B  e.  dom  F )  -> 
( F `  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   A.wral 2349   Ord word 4125   Oncon0 4126   dom cdm 4371   -->wf 4928   ` cfv 4932   Smo wsmo 5934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-tr 3884  df-id 4056  df-iord 4129  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940  df-smo 5935
This theorem is referenced by: (None)
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