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Theorem smoiso2 6623
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 5645 . . . . . . 7  |-  ( F : A -onto-> B  ->  F : A --> B )
2 smo11 6618 . . . . . . 7  |-  ( ( F : A --> B  /\  Smo  F )  ->  F : A -1-1-> B )
31, 2sylan 458 . . . . . 6  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  F : A -1-1-> B )
4 simpl 444 . . . . . 6  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  F : A -onto-> B )
5 df-f1o 5453 . . . . . 6  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
63, 4, 5sylanbrc 646 . . . . 5  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  F : A -1-1-onto-> B )
76adantl 453 . . . 4  |-  ( ( ( Ord  A  /\  B  C_  On )  /\  ( F : A -onto-> B  /\  Smo  F ) )  ->  F : A -1-1-onto-> B
)
8 fofn 5647 . . . . . 6  |-  ( F : A -onto-> B  ->  F  Fn  A )
9 smoord 6619 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
x  e.  y  <->  ( F `  x )  e.  ( F `  y ) ) )
10 epel 4489 . . . . . . . 8  |-  ( x  _E  y  <->  x  e.  y )
11 fvex 5734 . . . . . . . . 9  |-  ( F `
 y )  e. 
_V
1211epelc 4488 . . . . . . . 8  |-  ( ( F `  x )  _E  ( F `  y )  <->  ( F `  x )  e.  ( F `  y ) )
139, 10, 123bitr4g 280 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
x  _E  y  <->  ( F `  x )  _E  ( F `  y )
) )
1413ralrimivva 2790 . . . . . 6  |-  ( ( F  Fn  A  /\  Smo  F )  ->  A. x  e.  A  A. y  e.  A  ( x  _E  y  <->  ( F `  x )  _E  ( F `  y )
) )
158, 14sylan 458 . . . . 5  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  A. x  e.  A  A. y  e.  A  ( x  _E  y  <->  ( F `  x )  _E  ( F `  y ) ) )
1615adantl 453 . . . 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 5455 . . . 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 646 . . 3  |-  ( ( ( Ord  A  /\  B  C_  On )  /\  ( F : A -onto-> B  /\  Smo  F ) )  ->  F  Isom  _E  ,  _E  ( A ,  B
) )
1918ex 424 . 2  |-  ( ( Ord  A  /\  B  C_  On )  ->  (
( F : A -onto-> B  /\  Smo  F )  ->  F  Isom  _E  ,  _E  ( A ,  B
) ) )
20 isof1o 6037 . . . . . . 7  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A -1-1-onto-> B
)
21 f1ofo 5673 . . . . . . 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 978 . . . . 5  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  F : A -onto-> B )
24 smoiso 6616 . . . . 5  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  Smo  F )
2523, 24jca 519 . . . 4  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  -> 
( F : A -onto-> B  /\  Smo  F ) )
26253expib 1156 . . 3  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( Ord 
A  /\  B  C_  On )  ->  ( F : A -onto-> B  /\  Smo  F
) ) )
2726com12 29 . 2  |-  ( ( Ord  A  /\  B  C_  On )  ->  ( F  Isom  _E  ,  _E  ( A ,  B )  ->  ( F : A -onto-> B  /\  Smo  F
) ) )
2819, 27impbid 184 1  |-  ( ( Ord  A  /\  B  C_  On )  ->  (
( F : A -onto-> B  /\  Smo  F )  <-> 
F  Isom  _E  ,  _E  ( A ,  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1725   A.wral 2697    C_ wss 3312   class class class wbr 4204    _E cep 4484   Ord word 4572   Oncon0 4573    Fn wfn 5441   -->wf 5442   -1-1->wf1 5443   -onto->wfo 5444   -1-1-onto->wf1o 5445   ` cfv 5446    Isom wiso 5447   Smo wsmo 6599
This theorem is referenced by:  oismo  7501  cofsmo  8141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-smo 6600
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