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Theorem issmo2 6614
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 5602 . . . . 5  |-  ( ( F : A --> B  /\  B  C_  On )  ->  F : A --> On )
21ex 425 . . . 4  |-  ( F : A --> B  -> 
( B  C_  On  ->  F : A --> On ) )
3 fdm 5598 . . . . 5  |-  ( F : A --> B  ->  dom  F  =  A )
43feq2d 5584 . . . 4  |-  ( F : A --> B  -> 
( F : dom  F --> On  <->  F : A --> On ) )
52, 4sylibrd 227 . . 3  |-  ( F : A --> B  -> 
( B  C_  On  ->  F : dom  F --> On ) )
6 ordeq 4591 . . . . 5  |-  ( dom 
F  =  A  -> 
( Ord  dom  F  <->  Ord  A ) )
73, 6syl 16 . . . 4  |-  ( F : A --> B  -> 
( Ord  dom  F  <->  Ord  A ) )
87biimprd 216 . . 3  |-  ( F : A --> B  -> 
( Ord  A  ->  Ord 
dom  F ) )
93raleqdv 2912 . . . 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 216 . . 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 1262 . 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 6612 . 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 220 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 178    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   Ord word 4583   Oncon0 4584   dom cdm 4881   -->wf 5453   ` cfv 5457   Smo wsmo 6610
This theorem is referenced by:  alephsmo  7988  cofsmo  8154  cfsmolem  8155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-in 3329  df-ss 3336  df-uni 4018  df-tr 4306  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-fn 5460  df-f 5461  df-smo 6611
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