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Theorem smoiso2 6402
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 5467 . . . . . . 7  |-  ( F : A -onto-> B  ->  F : A --> B )
2 smo11 6397 . . . . . . 7  |-  ( ( F : A --> B  /\  Smo  F )  ->  F : A -1-1-> B )
31, 2sylan 457 . . . . . 6  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  F : A -1-1-> B )
4 simpl 443 . . . . . 6  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  F : A -onto-> B )
5 df-f1o 5278 . . . . . 6  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
63, 4, 5sylanbrc 645 . . . . 5  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  F : A -1-1-onto-> B )
76adantl 452 . . . 4  |-  ( ( ( Ord  A  /\  B  C_  On )  /\  ( F : A -onto-> B  /\  Smo  F ) )  ->  F : A -1-1-onto-> B
)
8 fofn 5469 . . . . . 6  |-  ( F : A -onto-> B  ->  F  Fn  A )
9 smoord 6398 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
x  e.  y  <->  ( F `  x )  e.  ( F `  y ) ) )
10 epel 4324 . . . . . . . 8  |-  ( x  _E  y  <->  x  e.  y )
11 fvex 5555 . . . . . . . . 9  |-  ( F `
 y )  e. 
_V
1211epelc 4323 . . . . . . . 8  |-  ( ( F `  x )  _E  ( F `  y )  <->  ( F `  x )  e.  ( F `  y ) )
139, 10, 123bitr4g 279 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  Smo  F )  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
x  _E  y  <->  ( F `  x )  _E  ( F `  y )
) )
1413ralrimivva 2648 . . . . . 6  |-  ( ( F  Fn  A  /\  Smo  F )  ->  A. x  e.  A  A. y  e.  A  ( x  _E  y  <->  ( F `  x )  _E  ( F `  y )
) )
158, 14sylan 457 . . . . 5  |-  ( ( F : A -onto-> B  /\  Smo  F )  ->  A. x  e.  A  A. y  e.  A  ( x  _E  y  <->  ( F `  x )  _E  ( F `  y ) ) )
1615adantl 452 . . . 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 5280 . . . 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 645 . . 3  |-  ( ( ( Ord  A  /\  B  C_  On )  /\  ( F : A -onto-> B  /\  Smo  F ) )  ->  F  Isom  _E  ,  _E  ( A ,  B
) )
1918ex 423 . 2  |-  ( ( Ord  A  /\  B  C_  On )  ->  (
( F : A -onto-> B  /\  Smo  F )  ->  F  Isom  _E  ,  _E  ( A ,  B
) ) )
20 isof1o 5838 . . . . . . 7  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A -1-1-onto-> B
)
21 f1ofo 5495 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
2220, 21syl 15 . . . . . 6  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  F : A -onto-> B )
23223ad2ant1 976 . . . . 5  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  F : A -onto-> B )
24 smoiso 6395 . . . . 5  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  ->  Smo  F )
2523, 24jca 518 . . . 4  |-  ( ( F  Isom  _E  ,  _E  ( A ,  B )  /\  Ord  A  /\  B  C_  On )  -> 
( F : A -onto-> B  /\  Smo  F ) )
26253expib 1154 . . 3  |-  ( F 
Isom  _E  ,  _E  ( A ,  B )  ->  ( ( Ord 
A  /\  B  C_  On )  ->  ( F : A -onto-> B  /\  Smo  F
) ) )
2726com12 27 . 2  |-  ( ( Ord  A  /\  B  C_  On )  ->  ( F  Isom  _E  ,  _E  ( A ,  B )  ->  ( F : A -onto-> B  /\  Smo  F
) ) )
2819, 27impbid 183 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 176    /\ wa 358    /\ w3a 934    e. wcel 1696   A.wral 2556    C_ wss 3165   class class class wbr 4039    _E cep 4319   Ord word 4407   Oncon0 4408    Fn wfn 5266   -->wf 5267   -1-1->wf1 5268   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272   Smo wsmo 6378
This theorem is referenced by:  oismo  7271  cofsmo  7911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-smo 6379
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