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Theorem smoiso 6265
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
StepHypRef Expression
1 isof1o 5674 . . . 4  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A -1-1-onto-> B
)
2 f1of 5329 . . . 4  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
31, 2syl 17 . . 3  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A --> B )
4 ffdm 5260 . . . . . 6  |-  ( F : A --> B  -> 
( F : dom  F --> B  /\  dom  F  C_  A ) )
54simpld 447 . . . . 5  |-  ( F : A --> B  ->  F : dom  F --> B )
6 fss 5254 . . . . 5  |-  ( ( F : dom  F --> B  /\  B  C_  On )  ->  F : dom  F --> On )
75, 6sylan 459 . . . 4  |-  ( ( F : A --> B  /\  B  C_  On )  ->  F : dom  F --> On )
873adant2 979 . . 3  |-  ( ( F : A --> B  /\  Ord  A  /\  B  C_  On )  ->  F : dom  F --> On )
93, 8syl3an1 1220 . 2  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  F : dom  F --> On )
10 fdm 5250 . . . . . . 7  |-  ( F : A --> B  ->  dom  F  =  A )
1110eqcomd 2258 . . . . . 6  |-  ( F : A --> B  ->  A  =  dom  F )
121, 2, 113syl 20 . . . . 5  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  A  =  dom  F )
13 ordeq 4292 . . . . 5  |-  ( A  =  dom  F  -> 
( Ord  A  <->  Ord  dom  F
) )
1412, 13syl 17 . . . 4  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( Ord  A  <->  Ord 
dom  F ) )
1514biimpa 472 . . 3  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A )  ->  Ord  dom  F )
16153adant3 980 . 2  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  Ord  dom  F )
1710eleq2d 2320 . . . . . . 7  |-  ( F : A --> B  -> 
( x  e.  dom  F  <-> 
x  e.  A ) )
1810eleq2d 2320 . . . . . . 7  |-  ( F : A --> B  -> 
( y  e.  dom  F  <-> 
y  e.  A ) )
1917, 18anbi12d 694 . . . . . 6  |-  ( F : A --> B  -> 
( ( x  e. 
dom  F  /\  y  e.  dom  F )  <->  ( x  e.  A  /\  y  e.  A ) ) )
201, 2, 193syl 20 . . . . 5  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( x  e.  dom  F  /\  y  e.  dom  F )  <-> 
( x  e.  A  /\  y  e.  A
) ) )
21 isorel 5675 . . . . . . . 8  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  _E  y  <->  ( F `  x )  _E  ( F `  y ) ) )
22 epel 4201 . . . . . . . 8  |-  ( x  _E  y  <->  x  e.  y )
23 fvex 5391 . . . . . . . . 9  |-  ( F `
 y )  e. 
_V
2423epelc 4200 . . . . . . . 8  |-  ( ( F `  x )  _E  ( F `  y )  <->  ( F `  x )  e.  ( F `  y ) )
2521, 22, 243bitr3g 280 . . . . . . 7  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  e.  y  <-> 
( F `  x
)  e.  ( F `
 y ) ) )
2625biimpd 200 . . . . . 6  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  e.  y  ->  ( F `  x )  e.  ( F `  y ) ) )
2726ex 425 . . . . 5  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( x  e.  A  /\  y  e.  A )  ->  (
x  e.  y  -> 
( F `  x
)  e.  ( F `
 y ) ) ) )
2820, 27sylbid 208 . . . 4  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( x  e.  dom  F  /\  y  e.  dom  F )  ->  ( x  e.  y  ->  ( F `  x )  e.  ( F `  y ) ) ) )
2928ralrimivv 2596 . . 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 981 . 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 6249 . 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 1141 1  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  Smo  F )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2509    C_ wss 3078   class class class wbr 3920    _E cep 4196   Ord word 4284   Oncon0 4285   dom cdm 4580   -->wf 4588   -1-1-onto->wf1o 4591   ` cfv 4592    Isom wiso 4593   Smo wsmo 6248
This theorem is referenced by:  smoiso2  6272
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fn 4603  df-f 4604  df-f1 4605  df-f1o 4607  df-fv 4608  df-isom 4609  df-smo 6249
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