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Theorem oismo 7271
Description: When  A is a subclass of  On,  F is a strictly monotone ordinal functions, and it is also complete (it is an isomorphism onto all of  A). The proof avoids ax-rep 4147 (the second statement is trivial under ax-rep 4147). (Contributed by Mario Carneiro, 26-Jun-2015.)
Hypothesis
Ref Expression
oismo.1  |-  F  = OrdIso
(  _E  ,  A
)
Assertion
Ref Expression
oismo  |-  ( A 
C_  On  ->  ( Smo 
F  /\  ran  F  =  A ) )

Proof of Theorem oismo
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 epweon 4591 . . . . . 6  |-  _E  We  On
2 wess 4396 . . . . . 6  |-  ( A 
C_  On  ->  (  _E  We  On  ->  _E  We  A ) )
31, 2mpi 16 . . . . 5  |-  ( A 
C_  On  ->  _E  We  A )
4 epse 4392 . . . . 5  |-  _E Se  A
5 oismo.1 . . . . . 6  |-  F  = OrdIso
(  _E  ,  A
)
65oiiso2 7262 . . . . 5  |-  ( (  _E  We  A  /\  _E Se  A )  ->  F  Isom  _E  ,  _E  ( dom  F ,  ran  F
) )
73, 4, 6sylancl 643 . . . 4  |-  ( A 
C_  On  ->  F  Isom  _E  ,  _E  ( dom 
F ,  ran  F
) )
85oicl 7260 . . . . 5  |-  Ord  dom  F
95oif 7261 . . . . . . 7  |-  F : dom  F --> A
10 frn 5411 . . . . . . 7  |-  ( F : dom  F --> A  ->  ran  F  C_  A )
119, 10ax-mp 8 . . . . . 6  |-  ran  F  C_  A
12 id 19 . . . . . 6  |-  ( A 
C_  On  ->  A  C_  On )
1311, 12syl5ss 3203 . . . . 5  |-  ( A 
C_  On  ->  ran  F  C_  On )
14 smoiso2 6402 . . . . 5  |-  ( ( Ord  dom  F  /\  ran  F  C_  On )  ->  ( ( F : dom  F -onto-> ran  F  /\  Smo  F )  <->  F  Isom  _E  ,  _E  ( dom  F ,  ran  F ) ) )
158, 13, 14sylancr 644 . . . 4  |-  ( A 
C_  On  ->  ( ( F : dom  F -onto-> ran  F  /\  Smo  F
)  <->  F  Isom  _E  ,  _E  ( dom  F ,  ran  F ) ) )
167, 15mpbird 223 . . 3  |-  ( A 
C_  On  ->  ( F : dom  F -onto-> ran  F  /\  Smo  F ) )
1716simprd 449 . 2  |-  ( A 
C_  On  ->  Smo  F
)
1811a1i 10 . . 3  |-  ( A 
C_  On  ->  ran  F  C_  A )
19 simprl 732 . . . . . . . 8  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  x  e.  A )
203adantr 451 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  _E  We  A )
214a1i 10 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  _E Se  A )
22 ffn 5405 . . . . . . . . . . . . . 14  |-  ( F : dom  F --> A  ->  F  Fn  dom  F )
239, 22mp1i 11 . . . . . . . . . . . . 13  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  F  Fn  dom  F )
24 simplrr 737 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  -.  x  e.  ran  F )
253ad2antrr 706 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  _E  We  A )
264a1i 10 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  _E Se  A )
27 simplrl 736 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  x  e.  A )
28 simpr 447 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
y  e.  dom  F
)
295oiiniseg 7264 . . . . . . . . . . . . . . . . . 18  |-  ( ( (  _E  We  A  /\  _E Se  A )  /\  ( x  e.  A  /\  y  e.  dom  F ) )  ->  (
( F `  y
)  _E  x  \/  x  e.  ran  F
) )
3025, 26, 27, 28, 29syl22anc 1183 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
( ( F `  y )  _E  x  \/  x  e.  ran  F ) )
3130ord 366 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
( -.  ( F `
 y )  _E  x  ->  x  e.  ran  F ) )
3224, 31mt3d 117 . . . . . . . . . . . . . . 15  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
( F `  y
)  _E  x )
33 vex 2804 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
3433epelc 4323 . . . . . . . . . . . . . . 15  |-  ( ( F `  y )  _E  x  <->  ( F `  y )  e.  x
)
3532, 34sylib 188 . . . . . . . . . . . . . 14  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
( F `  y
)  e.  x )
3635ralrimiva 2639 . . . . . . . . . . . . 13  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  A. y  e.  dom  F ( F `
 y )  e.  x )
37 ffnfv 5701 . . . . . . . . . . . . 13  |-  ( F : dom  F --> x  <->  ( F  Fn  dom  F  /\  A. y  e.  dom  F ( F `  y )  e.  x ) )
3823, 36, 37sylanbrc 645 . . . . . . . . . . . 12  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  F : dom  F --> x )
399, 22mp1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  F  Fn  dom  F )
4017ad2antrr 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  Smo  F )
41 smogt 6400 . . . . . . . . . . . . . . . . 17  |-  ( ( F  Fn  dom  F  /\  Smo  F  /\  y  e.  dom  F )  -> 
y  C_  ( F `  y ) )
4239, 40, 28, 41syl3anc 1182 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
y  C_  ( F `  y ) )
43 ordelon 4432 . . . . . . . . . . . . . . . . . 18  |-  ( ( Ord  dom  F  /\  y  e.  dom  F )  ->  y  e.  On )
448, 28, 43sylancr 644 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
y  e.  On )
45 simpll 730 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  A  C_  On )
4645, 27sseldd 3194 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  x  e.  On )
47 ontr2 4455 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  On  /\  x  e.  On )  ->  ( ( y  C_  ( F `  y )  /\  ( F `  y )  e.  x
)  ->  y  e.  x ) )
4844, 46, 47syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
( ( y  C_  ( F `  y )  /\  ( F `  y )  e.  x
)  ->  y  e.  x ) )
4942, 35, 48mp2and 660 . . . . . . . . . . . . . . 15  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
y  e.  x )
5049ex 423 . . . . . . . . . . . . . 14  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  (
y  e.  dom  F  ->  y  e.  x ) )
5150ssrdv 3198 . . . . . . . . . . . . 13  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  dom  F 
C_  x )
52 ssexg 4176 . . . . . . . . . . . . 13  |-  ( ( dom  F  C_  x  /\  x  e.  A
)  ->  dom  F  e. 
_V )
5351, 19, 52syl2anc 642 . . . . . . . . . . . 12  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  dom  F  e.  _V )
54 fex2 5417 . . . . . . . . . . . 12  |-  ( ( F : dom  F --> x  /\  dom  F  e. 
_V  /\  x  e.  A )  ->  F  e.  _V )
5538, 53, 19, 54syl3anc 1182 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  F  e.  _V )
565ordtype2 7265 . . . . . . . . . . 11  |-  ( (  _E  We  A  /\  _E Se  A  /\  F  e. 
_V )  ->  F  Isom  _E  ,  _E  ( dom  F ,  A ) )
5720, 21, 55, 56syl3anc 1182 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  F  Isom  _E  ,  _E  ( dom  F ,  A ) )
58 isof1o 5838 . . . . . . . . . 10  |-  ( F 
Isom  _E  ,  _E  ( dom  F ,  A
)  ->  F : dom  F -1-1-onto-> A )
5957, 58syl 15 . . . . . . . . 9  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  F : dom  F -1-1-onto-> A )
60 f1ofo 5495 . . . . . . . . 9  |-  ( F : dom  F -1-1-onto-> A  ->  F : dom  F -onto-> A
)
61 forn 5470 . . . . . . . . 9  |-  ( F : dom  F -onto-> A  ->  ran  F  =  A )
6259, 60, 613syl 18 . . . . . . . 8  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  ran  F  =  A )
6319, 62eleqtrrd 2373 . . . . . . 7  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  x  e.  ran  F )
6463expr 598 . . . . . 6  |-  ( ( A  C_  On  /\  x  e.  A )  ->  ( -.  x  e.  ran  F  ->  x  e.  ran  F ) )
6564pm2.18d 103 . . . . 5  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  ran  F )
6665ex 423 . . . 4  |-  ( A 
C_  On  ->  ( x  e.  A  ->  x  e.  ran  F ) )
6766ssrdv 3198 . . 3  |-  ( A 
C_  On  ->  A  C_  ran  F )
6818, 67eqssd 3209 . 2  |-  ( A 
C_  On  ->  ran  F  =  A )
6917, 68jca 518 1  |-  ( A 
C_  On  ->  ( Smo 
F  /\  ran  F  =  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    _E cep 4319   Se wse 4366    We wwe 4367   Ord word 4407   Oncon0 4408   dom cdm 4705   ran crn 4706    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272   Smo wsmo 6378  OrdIsocoi 7240
This theorem is referenced by:  oiid  7272  hsmexlem1  8068  hsmexlem2  8069
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  ax-un 4528
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-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  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-riota 6320  df-smo 6379  df-recs 6404  df-oi 7241
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