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Theorem smoiso2 7106
Description: The strictly monotone ordinal functions are also epsilon isomorphisms of subclasses of  On. (Contributed by Mario Carneiro, 20-Mar-2013.)
Assertion
Ref Expression
smoiso2  |-  ( ( Ord  A  /\  B  C_  On )  ->  (
( F : A -onto-> B  /\  Smo  F )  <-> 
F  Isom  _E  ,  _E  ( A ,  B ) ) )

Proof of Theorem smoiso2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fof 5806 . . . . . . 7  |-  ( F : A -onto-> B  ->  F : A --> B )
2 smo11 7101 . . . . . . 7  |-  ( ( F : A --> B  /\  Smo  F )  ->  F : A -1-1-> B )
31, 2sylan 479 . . . . . 6  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  F : A -1-1-> B )
4 simpl 464 . . . . . 6  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  F : A -onto-> B )
5 df-f1o 5596 . . . . . 6  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
63, 4, 5sylanbrc 677 . . . . 5  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  F : A -1-1-onto-> B )
76adantl 473 . . . 4  |-  ( ( ( Ord  A  /\  B  C_  On )  /\  ( F : A -onto-> B  /\  Smo  F ) )  ->  F : A -1-1-onto-> B
)
8 fofn 5808 . . . . . 6  |-  ( F : A -onto-> B  ->  F  Fn  A )
9 smoord 7102 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
x  e.  y  <->  ( F `  x )  e.  ( F `  y ) ) )
10 epel 4753 . . . . . . . 8  |-  ( x  _E  y  <->  x  e.  y )
11 fvex 5889 . . . . . . . . 9  |-  ( F `
 y )  e. 
_V
1211epelc 4752 . . . . . . . 8  |-  ( ( F `  x )  _E  ( F `  y )  <->  ( F `  x )  e.  ( F `  y ) )
139, 10, 123bitr4g 296 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
x  _E  y  <->  ( F `  x )  _E  ( F `  y )
) )
1413ralrimivva 2814 . . . . . 6  |-  ( ( F  Fn  A  /\  Smo  F )  ->  A. x  e.  A  A. y  e.  A  ( x  _E  y  <->  ( F `  x )  _E  ( F `  y )
) )
158, 14sylan 479 . . . . 5  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  A. x  e.  A  A. y  e.  A  ( x  _E  y  <->  ( F `  x )  _E  ( F `  y ) ) )
1615adantl 473 . . . 4  |-  ( ( ( Ord  A  /\  B  C_  On )  /\  ( F : A -onto-> B  /\  Smo  F ) )  ->  A. x  e.  A  A. y  e.  A  ( x  _E  y  <->  ( F `  x )  _E  ( F `  y ) ) )
17 df-isom 5598 . . . 4  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  <-> 
( F : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x  _E  y  <->  ( F `  x )  _E  ( F `  y ) ) ) )
187, 16, 17sylanbrc 677 . . 3  |-  ( ( ( Ord  A  /\  B  C_  On )  /\  ( F : A -onto-> B  /\  Smo  F ) )  ->  F  Isom  _E  ,  _E  ( A ,  B
) )
1918ex 441 . 2  |-  ( ( Ord  A  /\  B  C_  On )  ->  (
( F : A -onto-> B  /\  Smo  F )  ->  F  Isom  _E  ,  _E  ( A ,  B
) ) )
20 isof1o 6234 . . . . . . 7  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A -1-1-onto-> B
)
21 f1ofo 5835 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
2220, 21syl 17 . . . . . 6  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A -onto-> B )
23223ad2ant1 1051 . . . . 5  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  F : A -onto-> B )
24 smoiso 7099 . . . . 5  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  Smo  F )
2523, 24jca 541 . . . 4  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  -> 
( F : A -onto-> B  /\  Smo  F ) )
26253expib 1234 . . 3  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( Ord 
A  /\  B  C_  On )  ->  ( F : A -onto-> B  /\  Smo  F
) ) )
2726com12 31 . 2  |-  ( ( Ord  A  /\  B  C_  On )  ->  ( F  Isom  _E  ,  _E  ( A ,  B )  ->  ( F : A -onto-> B  /\  Smo  F
) ) )
2819, 27impbid 195 1  |-  ( ( Ord  A  /\  B  C_  On )  ->  (
( F : A -onto-> B  /\  Smo  F )  <-> 
F  Isom  _E  ,  _E  ( A ,  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    e. wcel 1904   A.wral 2756    C_ wss 3390   class class class wbr 4395    _E cep 4748   Ord word 5429   Oncon0 5430    Fn wfn 5584   -->wf 5585   -1-1->wf1 5586   -onto->wfo 5587   -1-1-onto->wf1o 5588   ` cfv 5589    Isom wiso 5590   Smo wsmo 7082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-ord 5433  df-on 5434  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-smo 7083
This theorem is referenced by:  oismo  8073  cofsmo  8717
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