MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  smodm2 Structured version   Unicode version

Theorem smodm2 7028
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 7024 . 2  |-  ( Smo 
F  ->  Ord  dom  F
)
2 fndm 5670 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
3 ordeq 4875 . . . 4  |-  ( dom 
F  =  A  -> 
( Ord  dom  F  <->  Ord  A ) )
42, 3syl 16 . . 3  |-  ( F  Fn  A  ->  ( Ord  dom  F  <->  Ord  A ) )
54biimpa 484 . 2  |-  ( ( F  Fn  A  /\  Ord  dom  F )  ->  Ord  A )
61, 5sylan2 474 1  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383   Ord word 4867   dom cdm 4989    Fn wfn 5573   Smo wsmo 7018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-in 3468  df-ss 3475  df-uni 4235  df-tr 4531  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-fn 5581  df-smo 7019
This theorem is referenced by:  smo11  7037  smoord  7038  smoword  7039  smogt  7040  smorndom  7041  coftr  8656
  Copyright terms: Public domain W3C validator