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Theorem smoiso 6379
Description: If  F is an isomorphism from an ordinal  A onto  B, which is a subset of the ordinals, then 
F is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)
Assertion
Ref Expression
smoiso  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  Smo  F )

Proof of Theorem smoiso
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isof1o 5822 . . . 4  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A -1-1-onto-> B
)
2 f1of 5472 . . . 4  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
31, 2syl 15 . . 3  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A --> B )
4 ffdm 5403 . . . . . 6  |-  ( F : A --> B  -> 
( F : dom  F --> B  /\  dom  F  C_  A ) )
54simpld 445 . . . . 5  |-  ( F : A --> B  ->  F : dom  F --> B )
6 fss 5397 . . . . 5  |-  ( ( F : dom  F --> B  /\  B  C_  On )  ->  F : dom  F --> On )
75, 6sylan 457 . . . 4  |-  ( ( F : A --> B  /\  B  C_  On )  ->  F : dom  F --> On )
873adant2 974 . . 3  |-  ( ( F : A --> B  /\  Ord  A  /\  B  C_  On )  ->  F : dom  F --> On )
93, 8syl3an1 1215 . 2  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  F : dom  F --> On )
10 fdm 5393 . . . . . . 7  |-  ( F : A --> B  ->  dom  F  =  A )
1110eqcomd 2288 . . . . . 6  |-  ( F : A --> B  ->  A  =  dom  F )
121, 2, 113syl 18 . . . . 5  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  A  =  dom  F )
13 ordeq 4399 . . . . 5  |-  ( A  =  dom  F  -> 
( Ord  A  <->  Ord  dom  F
) )
1412, 13syl 15 . . . 4  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( Ord  A  <->  Ord 
dom  F ) )
1514biimpa 470 . . 3  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A )  ->  Ord  dom  F )
16153adant3 975 . 2  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  Ord  dom  F )
1710eleq2d 2350 . . . . . . 7  |-  ( F : A --> B  -> 
( x  e.  dom  F  <-> 
x  e.  A ) )
1810eleq2d 2350 . . . . . . 7  |-  ( F : A --> B  -> 
( y  e.  dom  F  <-> 
y  e.  A ) )
1917, 18anbi12d 691 . . . . . 6  |-  ( F : A --> B  -> 
( ( x  e. 
dom  F  /\  y  e.  dom  F )  <->  ( x  e.  A  /\  y  e.  A ) ) )
201, 2, 193syl 18 . . . . 5  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( x  e.  dom  F  /\  y  e.  dom  F )  <-> 
( x  e.  A  /\  y  e.  A
) ) )
21 isorel 5823 . . . . . . . 8  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  _E  y  <->  ( F `  x )  _E  ( F `  y ) ) )
22 epel 4308 . . . . . . . 8  |-  ( x  _E  y  <->  x  e.  y )
23 fvex 5539 . . . . . . . . 9  |-  ( F `
 y )  e. 
_V
2423epelc 4307 . . . . . . . 8  |-  ( ( F `  x )  _E  ( F `  y )  <->  ( F `  x )  e.  ( F `  y ) )
2521, 22, 243bitr3g 278 . . . . . . 7  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  e.  y  <-> 
( F `  x
)  e.  ( F `
 y ) ) )
2625biimpd 198 . . . . . 6  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  e.  y  ->  ( F `  x )  e.  ( F `  y ) ) )
2726ex 423 . . . . 5  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( x  e.  A  /\  y  e.  A )  ->  (
x  e.  y  -> 
( F `  x
)  e.  ( F `
 y ) ) ) )
2820, 27sylbid 206 . . . 4  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( x  e.  dom  F  /\  y  e.  dom  F )  ->  ( x  e.  y  ->  ( F `  x )  e.  ( F `  y ) ) ) )
2928ralrimivv 2634 . . 3  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  A. x  e.  dom  F A. y  e.  dom  F ( x  e.  y  ->  ( F `  x )  e.  ( F `  y ) ) )
30293ad2ant1 976 . 2  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  A. x  e.  dom  F A. y  e.  dom  F ( x  e.  y  ->  ( F `  x )  e.  ( F `  y ) ) )
31 df-smo 6363 . 2  |-  ( Smo 
F  <->  ( F : dom  F --> On  /\  Ord  dom 
F  /\  A. x  e.  dom  F A. y  e.  dom  F ( x  e.  y  ->  ( F `  x )  e.  ( F `  y
) ) ) )
329, 16, 30, 31syl3anbrc 1136 1  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  Smo  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   class class class wbr 4023    _E cep 4303   Ord word 4391   Oncon0 4392   dom cdm 4689   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256   Smo wsmo 6362
This theorem is referenced by:  smoiso2  6386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-iota 5219  df-fn 5258  df-f 5259  df-f1 5260  df-f1o 5262  df-fv 5263  df-isom 5264  df-smo 6363
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