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Theorem smodm2 6372
Description: The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
smodm2  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )

Proof of Theorem smodm2
StepHypRef Expression
1 smodm 6368 . 2  |-  ( Smo 
F  ->  Ord  dom  F
)
2 fndm 5343 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
3 ordeq 4399 . . . 4  |-  ( dom 
F  =  A  -> 
( Ord  dom  F  <->  Ord  A ) )
42, 3syl 15 . . 3  |-  ( F  Fn  A  ->  ( Ord  dom  F  <->  Ord  A ) )
54biimpa 470 . 2  |-  ( ( F  Fn  A  /\  Ord  dom  F )  ->  Ord  A )
61, 5sylan2 460 1  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   Ord word 4391   dom cdm 4689    Fn wfn 5250   Smo wsmo 6362
This theorem is referenced by:  smo11  6381  smoord  6382  smoword  6383  smogt  6384  smorndom  6385  coftr  7899
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-in 3159  df-ss 3166  df-uni 3828  df-tr 4114  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-fn 5258  df-smo 6363
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