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Theorem smoiso 6167
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 5676 . . . 4  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A -1-1-onto-> B
)
2 f1of 5335 . . . 4  |-  ( F : A -1-1-onto-> B  ->  F : A
--> B )
31, 2syl 14 . . 3  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A --> B )
4 ffdm 5263 . . . . . 6  |-  ( F : A --> B  -> 
( F : dom  F --> B  /\  dom  F  C_  A ) )
54simpld 111 . . . . 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 281 . . . 4  |-  ( ( F : A --> B  /\  B  C_  On )  ->  F : dom  F --> On )
873adant2 985 . . 3  |-  ( ( F : A --> B  /\  Ord  A  /\  B  C_  On )  ->  F : dom  F --> On )
93, 8syl3an1 1234 . 2  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  F : dom  F --> On )
10 fdm 5248 . . . . . 6  |-  ( F : A --> B  ->  dom  F  =  A )
1110eqcomd 2123 . . . . 5  |-  ( F : A --> B  ->  A  =  dom  F )
12 ordeq 4264 . . . . 5  |-  ( A  =  dom  F  -> 
( Ord  A  <->  Ord  dom  F
) )
131, 2, 11, 124syl 18 . . . 4  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( Ord  A  <->  Ord 
dom  F ) )
1413biimpa 294 . . 3  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A )  ->  Ord  dom  F )
15143adant3 986 . 2  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  Ord  dom  F )
1610eleq2d 2187 . . . . . . 7  |-  ( F : A --> B  -> 
( x  e.  dom  F  <-> 
x  e.  A ) )
1710eleq2d 2187 . . . . . . 7  |-  ( F : A --> B  -> 
( y  e.  dom  F  <-> 
y  e.  A ) )
1816, 17anbi12d 464 . . . . . 6  |-  ( F : A --> B  -> 
( ( x  e. 
dom  F  /\  y  e.  dom  F )  <->  ( x  e.  A  /\  y  e.  A ) ) )
191, 2, 183syl 17 . . . . 5  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( x  e.  dom  F  /\  y  e.  dom  F )  <-> 
( x  e.  A  /\  y  e.  A
) ) )
20 epel 4184 . . . . . . . . 9  |-  ( x  _E  y  <->  x  e.  y )
21 isorel 5677 . . . . . . . . 9  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  _E  y  <->  ( F `  x )  _E  ( F `  y ) ) )
2220, 21syl5bbr 193 . . . . . . . 8  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  e.  y  <-> 
( F `  x
)  _E  ( F `
 y ) ) )
23 ffn 5242 . . . . . . . . . . 11  |-  ( F : A --> B  ->  F  Fn  A )
243, 23syl 14 . . . . . . . . . 10  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F  Fn  A
)
2524adantr 274 . . . . . . . . 9  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  ->  F  Fn  A )
26 simprr 506 . . . . . . . . 9  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
y  e.  A )
27 funfvex 5406 . . . . . . . . . . 11  |-  ( ( Fun  F  /\  y  e.  dom  F )  -> 
( F `  y
)  e.  _V )
2827funfni 5193 . . . . . . . . . 10  |-  ( ( F  Fn  A  /\  y  e.  A )  ->  ( F `  y
)  e.  _V )
29 epelg 4182 . . . . . . . . . 10  |-  ( ( F `  y )  e.  _V  ->  (
( F `  x
)  _E  ( F `
 y )  <->  ( F `  x )  e.  ( F `  y ) ) )
3028, 29syl 14 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  y  e.  A )  ->  ( ( F `  x )  _E  ( F `  y )  <->  ( F `  x )  e.  ( F `  y ) ) )
3125, 26, 30syl2anc 408 . . . . . . . 8  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( ( F `  x )  _E  ( F `  y )  <->  ( F `  x )  e.  ( F `  y ) ) )
3222, 31bitrd 187 . . . . . . 7  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  e.  y  <-> 
( F `  x
)  e.  ( F `
 y ) ) )
3332biimpd 143 . . . . . 6  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  e.  y  ->  ( F `  x )  e.  ( F `  y ) ) )
3433ex 114 . . . . 5  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( x  e.  A  /\  y  e.  A )  ->  (
x  e.  y  -> 
( F `  x
)  e.  ( F `
 y ) ) ) )
3519, 34sylbid 149 . . . 4  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( x  e.  dom  F  /\  y  e.  dom  F )  ->  ( x  e.  y  ->  ( F `  x )  e.  ( F `  y ) ) ) )
3635ralrimivv 2490 . . 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 ) ) )
37363ad2ant1 987 . 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 ) ) )
38 df-smo 6151 . 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
) ) ) )
399, 15, 37, 38syl3anbrc 1150 1  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  Smo  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   A.wral 2393   _Vcvv 2660    C_ wss 3041   class class class wbr 3899    _E cep 4179   Ord word 4254   Oncon0 4255   dom cdm 4509    Fn wfn 5088   -->wf 5089   -1-1-onto->wf1o 5092   ` cfv 5093    Isom wiso 5094   Smo wsmo 6150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-tr 3997  df-eprel 4181  df-id 4185  df-iord 4258  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-f1o 5100  df-fv 5101  df-isom 5102  df-smo 6151
This theorem is referenced by: (None)
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