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Theorem smoiso 6067
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 5586 . . . 4  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A -1-1-onto-> B
)
2 f1of 5253 . . . 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 5181 . . . . . 6  |-  ( F : A --> B  -> 
( F : dom  F --> B  /\  dom  F  C_  A ) )
54simpld 110 . . . . 5  |-  ( F : A --> B  ->  F : dom  F --> B )
6 fss 5172 . . . . 5  |-  ( ( F : dom  F --> B  /\  B  C_  On )  ->  F : dom  F --> On )
75, 6sylan 277 . . . 4  |-  ( ( F : A --> B  /\  B  C_  On )  ->  F : dom  F --> On )
873adant2 962 . . 3  |-  ( ( F : A --> B  /\  Ord  A  /\  B  C_  On )  ->  F : dom  F --> On )
93, 8syl3an1 1207 . 2  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  F : dom  F --> On )
10 fdm 5166 . . . . . 6  |-  ( F : A --> B  ->  dom  F  =  A )
1110eqcomd 2093 . . . . 5  |-  ( F : A --> B  ->  A  =  dom  F )
12 ordeq 4199 . . . . 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 290 . . 3  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A )  ->  Ord  dom  F )
15143adant3 963 . 2  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  Ord  dom  F )
1610eleq2d 2157 . . . . . . 7  |-  ( F : A --> B  -> 
( x  e.  dom  F  <-> 
x  e.  A ) )
1710eleq2d 2157 . . . . . . 7  |-  ( F : A --> B  -> 
( y  e.  dom  F  <-> 
y  e.  A ) )
1816, 17anbi12d 457 . . . . . 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 4119 . . . . . . . . 9  |-  ( x  _E  y  <->  x  e.  y )
21 isorel 5587 . . . . . . . . 9  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  _E  y  <->  ( F `  x )  _E  ( F `  y ) ) )
2220, 21syl5bbr 192 . . . . . . . 8  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  e.  y  <-> 
( F `  x
)  _E  ( F `
 y ) ) )
23 ffn 5161 . . . . . . . . . . 11  |-  ( F : A --> B  ->  F  Fn  A )
243, 23syl 14 . . . . . . . . . 10  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F  Fn  A
)
2524adantr 270 . . . . . . . . 9  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  ->  F  Fn  A )
26 simprr 499 . . . . . . . . 9  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
y  e.  A )
27 funfvex 5322 . . . . . . . . . . 11  |-  ( ( Fun  F  /\  y  e.  dom  F )  -> 
( F `  y
)  e.  _V )
2827funfni 5114 . . . . . . . . . 10  |-  ( ( F  Fn  A  /\  y  e.  A )  ->  ( F `  y
)  e.  _V )
29 epelg 4117 . . . . . . . . . 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 403 . . . . . . . 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 186 . . . . . . 7  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  e.  y  <-> 
( F `  x
)  e.  ( F `
 y ) ) )
3332biimpd 142 . . . . . 6  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  e.  y  ->  ( F `  x )  e.  ( F `  y ) ) )
3433ex 113 . . . . 5  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( x  e.  A  /\  y  e.  A )  ->  (
x  e.  y  -> 
( F `  x
)  e.  ( F `
 y ) ) ) )
3519, 34sylbid 148 . . . 4  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( x  e.  dom  F  /\  y  e.  dom  F )  ->  ( x  e.  y  ->  ( F `  x )  e.  ( F `  y ) ) ) )
3635ralrimivv 2454 . . 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 964 . 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 6051 . 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 1127 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 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   A.wral 2359   _Vcvv 2619    C_ wss 2999   class class class wbr 3845    _E cep 4114   Ord word 4189   Oncon0 4190   dom cdm 4438    Fn wfn 5010   -->wf 5011   -1-1-onto->wf1o 5014   ` cfv 5015    Isom wiso 5016   Smo wsmo 6050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-tr 3937  df-eprel 4116  df-id 4120  df-iord 4193  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-f1o 5022  df-fv 5023  df-isom 5024  df-smo 6051
This theorem is referenced by: (None)
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