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Definition df-smo 7073
Description: Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.)
Assertion
Ref Expression
df-smo  |-  ( Smo 
A  <->  ( A : dom  A --> On  /\  Ord  dom 
A  /\  A. x  e.  dom  A A. y  e.  dom  A ( x  e.  y  ->  ( A `  x )  e.  ( A `  y
) ) ) )
Distinct variable group:    x, y, A

Detailed syntax breakdown of Definition df-smo
StepHypRef Expression
1 cA . . 3  class  A
21wsmo 7072 . 2  wff  Smo  A
31cdm 4854 . . . 4  class  dom  A
4 con0 5442 . . . 4  class  On
53, 4, 1wf 5597 . . 3  wff  A : dom  A --> On
63word 5441 . . 3  wff  Ord  dom  A
7 vx . . . . . . 7  setvar  x
8 vy . . . . . . 7  setvar  y
97, 8wel 1871 . . . . . 6  wff  x  e.  y
107cv 1436 . . . . . . . 8  class  x
1110, 1cfv 5601 . . . . . . 7  class  ( A `
 x )
128cv 1436 . . . . . . . 8  class  y
1312, 1cfv 5601 . . . . . . 7  class  ( A `
 y )
1411, 13wcel 1870 . . . . . 6  wff  ( A `
 x )  e.  ( A `  y
)
159, 14wi 4 . . . . 5  wff  ( x  e.  y  ->  ( A `  x )  e.  ( A `  y
) )
1615, 8, 3wral 2782 . . . 4  wff  A. y  e.  dom  A ( x  e.  y  ->  ( A `  x )  e.  ( A `  y
) )
1716, 7, 3wral 2782 . . 3  wff  A. x  e.  dom  A A. y  e.  dom  A ( x  e.  y  ->  ( A `  x )  e.  ( A `  y
) )
185, 6, 17w3a 982 . 2  wff  ( A : dom  A --> On  /\  Ord  dom  A  /\  A. x  e.  dom  A A. y  e.  dom  A ( x  e.  y  -> 
( A `  x
)  e.  ( A `
 y ) ) )
192, 18wb 187 1  wff  ( Smo 
A  <->  ( A : dom  A --> On  /\  Ord  dom 
A  /\  A. x  e.  dom  A A. y  e.  dom  A ( x  e.  y  ->  ( A `  x )  e.  ( A `  y
) ) ) )
Colors of variables: wff setvar class
This definition is referenced by:  dfsmo2  7074  issmo  7075  smoeq  7077  smodm  7078  smores  7079  smofvon  7086  smoel  7087  smoiso  7089
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