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Theorem smodm2 6388
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 6384 . 2  |-  ( Smo 
F  ->  Ord  dom  F
)
2 fndm 5359 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
3 ordeq 4415 . . . 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 1632   Ord word 4407   dom cdm 4705    Fn wfn 5266   Smo wsmo 6378
This theorem is referenced by:  smo11  6397  smoord  6398  smoword  6399  smogt  6400  smorndom  6401  coftr  7915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-in 3172  df-ss 3179  df-uni 3844  df-tr 4130  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-fn 5274  df-smo 6379
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