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Theorem smofvon2dm 5911
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 5902 . . 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 919 . 2  |-  ( Smo 
F  ->  F : dom  F --> On )
32ffvelrnda 5302 1  |-  ( ( Smo  F  /\  B  e.  dom  F )  -> 
( F `  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    e. wcel 1393   A.wral 2306   Ord word 4099   Oncon0 4100   dom cdm 4345   -->wf 4898   ` cfv 4902   Smo wsmo 5900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-tr 3855  df-id 4030  df-iord 4103  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-smo 5901
This theorem is referenced by: (None)
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