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Theorem smoiso2 6568
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 5594 . . . . . . 7  |-  ( F : A -onto-> B  ->  F : A --> B )
2 smo11 6563 . . . . . . 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 5402 . . . . . 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 5596 . . . . . 6  |-  ( F : A -onto-> B  ->  F  Fn  A )
9 smoord 6564 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
x  e.  y  <->  ( F `  x )  e.  ( F `  y ) ) )
10 epel 4439 . . . . . . . 8  |-  ( x  _E  y  <->  x  e.  y )
11 fvex 5683 . . . . . . . . 9  |-  ( F `
 y )  e. 
_V
1211epelc 4438 . . . . . . . 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 2742 . . . . . 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 5404 . . . 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 5985 . . . . . . 7  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A -1-1-onto-> B
)
21 f1ofo 5622 . . . . . . 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 6561 . . . . 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 1717   A.wral 2650    C_ wss 3264   class class class wbr 4154    _E cep 4434   Ord word 4522   Oncon0 4523    Fn wfn 5390   -->wf 5391   -1-1->wf1 5392   -onto->wfo 5393   -1-1-onto->wf1o 5394   ` cfv 5395    Isom wiso 5396   Smo wsmo 6544
This theorem is referenced by:  oismo  7443  cofsmo  8083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-smo 6545
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