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Theorem issmo2 6910
Description: Alternative definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
issmo2  |-  ( F : A --> B  -> 
( ( B  C_  On  /\  Ord  A  /\  A. x  e.  A  A. y  e.  x  ( F `  y )  e.  ( F `  x
) )  ->  Smo  F ) )
Distinct variable groups:    x, A    x, F, y
Allowed substitution hints:    A( y)    B( x, y)

Proof of Theorem issmo2
StepHypRef Expression
1 fss 5665 . . . . 5  |-  ( ( F : A --> B  /\  B  C_  On )  ->  F : A --> On )
21ex 434 . . . 4  |-  ( F : A --> B  -> 
( B  C_  On  ->  F : A --> On ) )
3 fdm 5661 . . . . 5  |-  ( F : A --> B  ->  dom  F  =  A )
43feq2d 5645 . . . 4  |-  ( F : A --> B  -> 
( F : dom  F --> On  <->  F : A --> On ) )
52, 4sylibrd 234 . . 3  |-  ( F : A --> B  -> 
( B  C_  On  ->  F : dom  F --> On ) )
6 ordeq 4824 . . . . 5  |-  ( dom 
F  =  A  -> 
( Ord  dom  F  <->  Ord  A ) )
73, 6syl 16 . . . 4  |-  ( F : A --> B  -> 
( Ord  dom  F  <->  Ord  A ) )
87biimprd 223 . . 3  |-  ( F : A --> B  -> 
( Ord  A  ->  Ord 
dom  F ) )
93raleqdv 3019 . . . 4  |-  ( F : A --> B  -> 
( A. x  e. 
dom  F A. y  e.  x  ( F `  y )  e.  ( F `  x )  <->  A. x  e.  A  A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
109biimprd 223 . . 3  |-  ( F : A --> B  -> 
( A. x  e.  A  A. y  e.  x  ( F `  y )  e.  ( F `  x )  ->  A. x  e.  dom  F A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
115, 8, 103anim123d 1297 . 2  |-  ( F : A --> B  -> 
( ( B  C_  On  /\  Ord  A  /\  A. x  e.  A  A. y  e.  x  ( F `  y )  e.  ( F `  x
) )  ->  ( F : dom  F --> On  /\  Ord  dom  F  /\  A. x  e.  dom  F A. y  e.  x  ( F `  y )  e.  ( F `  x
) ) ) )
12 dfsmo2 6908 . 2  |-  ( Smo 
F  <->  ( F : dom  F --> On  /\  Ord  dom 
F  /\  A. x  e.  dom  F A. y  e.  x  ( F `  y )  e.  ( F `  x ) ) )
1311, 12syl6ibr 227 1  |-  ( F : A --> B  -> 
( ( B  C_  On  /\  Ord  A  /\  A. x  e.  A  A. y  e.  x  ( F `  y )  e.  ( F `  x
) )  ->  Smo  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795    C_ wss 3426   Ord word 4816   Oncon0 4817   dom cdm 4938   -->wf 5512   ` cfv 5516   Smo wsmo 6906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-v 3070  df-in 3433  df-ss 3440  df-uni 4190  df-tr 4484  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-fn 5519  df-f 5520  df-smo 6907
This theorem is referenced by:  alephsmo  8373  cofsmo  8539  cfsmolem  8540
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