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Theorem smodm2 6554
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 6550 . 2  |-  ( Smo 
F  ->  Ord  dom  F
)
2 fndm 5485 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
3 ordeq 4530 . . . 4  |-  ( dom 
F  =  A  -> 
( Ord  dom  F  <->  Ord  A ) )
42, 3syl 16 . . 3  |-  ( F  Fn  A  ->  ( Ord  dom  F  <->  Ord  A ) )
54biimpa 471 . 2  |-  ( ( F  Fn  A  /\  Ord  dom  F )  ->  Ord  A )
61, 5sylan2 461 1  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649   Ord word 4522   dom cdm 4819    Fn wfn 5390   Smo wsmo 6544
This theorem is referenced by:  smo11  6563  smoord  6564  smoword  6565  smogt  6566  smorndom  6567  coftr  8087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-in 3271  df-ss 3278  df-uni 3959  df-tr 4245  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-fn 5398  df-smo 6545
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