ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  smoiso Unicode version

Theorem smoiso 6297
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 5802 . . . 4  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A -1-1-onto-> B
)
2 f1of 5457 . . . 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 5382 . . . . . 6  |-  ( F : A --> B  -> 
( F : dom  F --> B  /\  dom  F  C_  A ) )
54simpld 112 . . . . 5  |-  ( F : A --> B  ->  F : dom  F --> B )
6 fss 5373 . . . . 5  |-  ( ( F : dom  F --> B  /\  B  C_  On )  ->  F : dom  F --> On )
75, 6sylan 283 . . . 4  |-  ( ( F : A --> B  /\  B  C_  On )  ->  F : dom  F --> On )
873adant2 1016 . . 3  |-  ( ( F : A --> B  /\  Ord  A  /\  B  C_  On )  ->  F : dom  F --> On )
93, 8syl3an1 1271 . 2  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  F : dom  F --> On )
10 fdm 5367 . . . . . 6  |-  ( F : A --> B  ->  dom  F  =  A )
1110eqcomd 2183 . . . . 5  |-  ( F : A --> B  ->  A  =  dom  F )
12 ordeq 4369 . . . . 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 296 . . 3  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A )  ->  Ord  dom  F )
15143adant3 1017 . 2  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  Ord  dom  F )
1610eleq2d 2247 . . . . . . 7  |-  ( F : A --> B  -> 
( x  e.  dom  F  <-> 
x  e.  A ) )
1710eleq2d 2247 . . . . . . 7  |-  ( F : A --> B  -> 
( y  e.  dom  F  <-> 
y  e.  A ) )
1816, 17anbi12d 473 . . . . . 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 4289 . . . . . . . . 9  |-  ( x  _E  y  <->  x  e.  y )
21 isorel 5803 . . . . . . . . 9  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  _E  y  <->  ( F `  x )  _E  ( F `  y ) ) )
2220, 21bitr3id 194 . . . . . . . 8  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  e.  y  <-> 
( F `  x
)  _E  ( F `
 y ) ) )
23 ffn 5361 . . . . . . . . . . 11  |-  ( F : A --> B  ->  F  Fn  A )
243, 23syl 14 . . . . . . . . . 10  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F  Fn  A
)
2524adantr 276 . . . . . . . . 9  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  ->  F  Fn  A )
26 simprr 531 . . . . . . . . 9  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
y  e.  A )
27 funfvex 5528 . . . . . . . . . . 11  |-  ( ( Fun  F  /\  y  e.  dom  F )  -> 
( F `  y
)  e.  _V )
2827funfni 5312 . . . . . . . . . 10  |-  ( ( F  Fn  A  /\  y  e.  A )  ->  ( F `  y
)  e.  _V )
29 epelg 4287 . . . . . . . . . 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 411 . . . . . . . 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 188 . . . . . . 7  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  e.  y  <-> 
( F `  x
)  e.  ( F `
 y ) ) )
3332biimpd 144 . . . . . 6  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  e.  y  ->  ( F `  x )  e.  ( F `  y ) ) )
3433ex 115 . . . . 5  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( x  e.  A  /\  y  e.  A )  ->  (
x  e.  y  -> 
( F `  x
)  e.  ( F `
 y ) ) ) )
3519, 34sylbid 150 . . . 4  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( x  e.  dom  F  /\  y  e.  dom  F )  ->  ( x  e.  y  ->  ( F `  x )  e.  ( F `  y ) ) ) )
3635ralrimivv 2558 . . 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 1018 . 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 6281 . 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 1181 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 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   _Vcvv 2737    C_ wss 3129   class class class wbr 4000    _E cep 4284   Ord word 4359   Oncon0 4360   dom cdm 4623    Fn wfn 5207   -->wf 5208   -1-1-onto->wf1o 5211   ` cfv 5212    Isom wiso 5213   Smo wsmo 6280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-tr 4099  df-eprel 4286  df-id 4290  df-iord 4363  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-f1o 5219  df-fv 5220  df-isom 5221  df-smo 6281
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator