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Theorem issmo2 7012
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 5721 . . . . 5  |-  ( ( F : A --> B  /\  B  C_  On )  ->  F : A --> On )
21ex 432 . . . 4  |-  ( F : A --> B  -> 
( B  C_  On  ->  F : A --> On ) )
3 fdm 5717 . . . . 5  |-  ( F : A --> B  ->  dom  F  =  A )
43feq2d 5700 . . . 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 4874 . . . . 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 3057 . . . 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 1304 . 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 7010 . 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 971    = wceq 1398    e. wcel 1823   A.wral 2804    C_ wss 3461   Ord word 4866   Oncon0 4867   dom cdm 4988   -->wf 5566   ` cfv 5570   Smo wsmo 7008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-v 3108  df-in 3468  df-ss 3475  df-uni 4236  df-tr 4533  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-fn 5573  df-f 5574  df-smo 7009
This theorem is referenced by:  alephsmo  8474  cofsmo  8640  cfsmolem  8641
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