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Theorem issmo2 6574
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 5562 . . . . 5  |-  ( ( F : A --> B  /\  B  C_  On )  ->  F : A --> On )
21ex 424 . . . 4  |-  ( F : A --> B  -> 
( B  C_  On  ->  F : A --> On ) )
3 fdm 5558 . . . . 5  |-  ( F : A --> B  ->  dom  F  =  A )
43feq2d 5544 . . . 4  |-  ( F : A --> B  -> 
( F : dom  F --> On  <->  F : A --> On ) )
52, 4sylibrd 226 . . 3  |-  ( F : A --> B  -> 
( B  C_  On  ->  F : dom  F --> On ) )
6 ordeq 4552 . . . . 5  |-  ( dom 
F  =  A  -> 
( Ord  dom  F  <->  Ord  A ) )
73, 6syl 16 . . . 4  |-  ( F : A --> B  -> 
( Ord  dom  F  <->  Ord  A ) )
87biimprd 215 . . 3  |-  ( F : A --> B  -> 
( Ord  A  ->  Ord 
dom  F ) )
93raleqdv 2874 . . . 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 215 . . 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 1261 . 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 6572 . 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 219 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670    C_ wss 3284   Ord word 4544   Oncon0 4545   dom cdm 4841   -->wf 5413   ` cfv 5417   Smo wsmo 6570
This theorem is referenced by:  alephsmo  7943  cofsmo  8109  cfsmolem  8110
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-v 2922  df-in 3291  df-ss 3298  df-uni 3980  df-tr 4267  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-fn 5420  df-f 5421  df-smo 6571
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