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Theorem issmo2 5904
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 5054 . . . . 5  |-  ( ( F : A --> B  /\  B  C_  On )  ->  F : A --> On )
21ex 108 . . . 4  |-  ( F : A --> B  -> 
( B  C_  On  ->  F : A --> On ) )
3 fdm 5050 . . . . 5  |-  ( F : A --> B  ->  dom  F  =  A )
43feq2d 5035 . . . 4  |-  ( F : A --> B  -> 
( F : dom  F --> On  <->  F : A --> On ) )
52, 4sylibrd 158 . . 3  |-  ( F : A --> B  -> 
( B  C_  On  ->  F : dom  F --> On ) )
6 ordeq 4109 . . . . 5  |-  ( dom 
F  =  A  -> 
( Ord  dom  F  <->  Ord  A ) )
73, 6syl 14 . . . 4  |-  ( F : A --> B  -> 
( Ord  dom  F  <->  Ord  A ) )
87biimprd 147 . . 3  |-  ( F : A --> B  -> 
( Ord  A  ->  Ord 
dom  F ) )
93raleqdv 2511 . . . 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 147 . . 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 1214 . 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 5902 . 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 151 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 98    /\ w3a 885    = wceq 1243    e. wcel 1393   A.wral 2306    C_ wss 2917   Ord word 4099   Oncon0 4100   dom cdm 4345   -->wf 4898   ` cfv 4902   Smo wsmo 5900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-fn 4905  df-f 4906  df-smo 5901
This theorem is referenced by: (None)
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