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Theorem smoiso2 7041
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 5795 . . . . . . 7  |-  ( F : A -onto-> B  ->  F : A --> B )
2 smo11 7036 . . . . . . 7  |-  ( ( F : A --> B  /\  Smo  F )  ->  F : A -1-1-> B )
31, 2sylan 471 . . . . . 6  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  F : A -1-1-> B )
4 simpl 457 . . . . . 6  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  F : A -onto-> B )
5 df-f1o 5595 . . . . . 6  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
63, 4, 5sylanbrc 664 . . . . 5  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  F : A -1-1-onto-> B )
76adantl 466 . . . 4  |-  ( ( ( Ord  A  /\  B  C_  On )  /\  ( F : A -onto-> B  /\  Smo  F ) )  ->  F : A -1-1-onto-> B
)
8 fofn 5797 . . . . . 6  |-  ( F : A -onto-> B  ->  F  Fn  A )
9 smoord 7037 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
x  e.  y  <->  ( F `  x )  e.  ( F `  y ) ) )
10 epel 4794 . . . . . . . 8  |-  ( x  _E  y  <->  x  e.  y )
11 fvex 5876 . . . . . . . . 9  |-  ( F `
 y )  e. 
_V
1211epelc 4793 . . . . . . . 8  |-  ( ( F `  x )  _E  ( F `  y )  <->  ( F `  x )  e.  ( F `  y ) )
139, 10, 123bitr4g 288 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
x  _E  y  <->  ( F `  x )  _E  ( F `  y )
) )
1413ralrimivva 2885 . . . . . 6  |-  ( ( F  Fn  A  /\  Smo  F )  ->  A. x  e.  A  A. y  e.  A  ( x  _E  y  <->  ( F `  x )  _E  ( F `  y )
) )
158, 14sylan 471 . . . . 5  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  A. x  e.  A  A. y  e.  A  ( x  _E  y  <->  ( F `  x )  _E  ( F `  y ) ) )
1615adantl 466 . . . 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 5597 . . . 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 664 . . 3  |-  ( ( ( Ord  A  /\  B  C_  On )  /\  ( F : A -onto-> B  /\  Smo  F ) )  ->  F  Isom  _E  ,  _E  ( A ,  B
) )
1918ex 434 . 2  |-  ( ( Ord  A  /\  B  C_  On )  ->  (
( F : A -onto-> B  /\  Smo  F )  ->  F  Isom  _E  ,  _E  ( A ,  B
) ) )
20 isof1o 6210 . . . . . . 7  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A -1-1-onto-> B
)
21 f1ofo 5823 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
2220, 21syl 16 . . . . . 6  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A -onto-> B )
23223ad2ant1 1017 . . . . 5  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  F : A -onto-> B )
24 smoiso 7034 . . . . 5  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  Smo  F )
2523, 24jca 532 . . . 4  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  -> 
( F : A -onto-> B  /\  Smo  F ) )
26253expib 1199 . . 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 191 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 184    /\ wa 369    /\ w3a 973    e. wcel 1767   A.wral 2814    C_ wss 3476   class class class wbr 4447    _E cep 4789   Ord word 4877   Oncon0 4878    Fn wfn 5583   -->wf 5584   -1-1->wf1 5585   -onto->wfo 5586   -1-1-onto->wf1o 5587   ` cfv 5588    Isom wiso 5589   Smo wsmo 7017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-smo 7018
This theorem is referenced by:  oismo  7966  cofsmo  8650
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