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Theorem oismo 7538
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 4345 (the second statement is trivial under ax-rep 4345). (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 4793 . . . . . 6  |-  _E  We  On
2 wess 4598 . . . . . 6  |-  ( A 
C_  On  ->  (  _E  We  On  ->  _E  We  A ) )
31, 2mpi 17 . . . . 5  |-  ( A 
C_  On  ->  _E  We  A )
4 epse 4594 . . . . 5  |-  _E Se  A
5 oismo.1 . . . . . 6  |-  F  = OrdIso
(  _E  ,  A
)
65oiiso2 7529 . . . . 5  |-  ( (  _E  We  A  /\  _E Se  A )  ->  F  Isom  _E  ,  _E  ( dom  F ,  ran  F
) )
73, 4, 6sylancl 645 . . . 4  |-  ( A 
C_  On  ->  F  Isom  _E  ,  _E  ( dom 
F ,  ran  F
) )
85oicl 7527 . . . . 5  |-  Ord  dom  F
95oif 7528 . . . . . . 7  |-  F : dom  F --> A
10 frn 5626 . . . . . . 7  |-  ( F : dom  F --> A  ->  ran  F  C_  A )
119, 10ax-mp 5 . . . . . 6  |-  ran  F  C_  A
12 id 21 . . . . . 6  |-  ( A 
C_  On  ->  A  C_  On )
1311, 12syl5ss 3345 . . . . 5  |-  ( A 
C_  On  ->  ran  F  C_  On )
14 smoiso2 6660 . . . . 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 646 . . . 4  |-  ( A 
C_  On  ->  ( ( F : dom  F -onto-> ran  F  /\  Smo  F
)  <->  F  Isom  _E  ,  _E  ( dom  F ,  ran  F ) ) )
167, 15mpbird 225 . . 3  |-  ( A 
C_  On  ->  ( F : dom  F -onto-> ran  F  /\  Smo  F ) )
1716simprd 451 . 2  |-  ( A 
C_  On  ->  Smo  F
)
1811a1i 11 . . 3  |-  ( A 
C_  On  ->  ran  F  C_  A )
19 simprl 734 . . . . . . . 8  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  x  e.  A )
203adantr 453 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  _E  We  A )
214a1i 11 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  _E Se  A )
22 ffn 5620 . . . . . . . . . . . . 13  |-  ( F : dom  F --> A  ->  F  Fn  dom  F )
239, 22mp1i 12 . . . . . . . . . . . 12  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  F  Fn  dom  F )
24 simplrr 739 . . . . . . . . . . . . . . 15  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  -.  x  e.  ran  F )
253ad2antrr 708 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  _E  We  A )
264a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  _E Se  A )
27 simplrl 738 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  x  e.  A )
28 simpr 449 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
y  e.  dom  F
)
295oiiniseg 7531 . . . . . . . . . . . . . . . . 17  |-  ( ( (  _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 1186 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
( ( F `  y )  _E  x  \/  x  e.  ran  F ) )
3130ord 368 . . . . . . . . . . . . . . 15  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
( -.  ( F `
 y )  _E  x  ->  x  e.  ran  F ) )
3224, 31mt3d 120 . . . . . . . . . . . . . 14  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
( F `  y
)  _E  x )
33 vex 2965 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
3433epelc 4525 . . . . . . . . . . . . . 14  |-  ( ( F `  y )  _E  x  <->  ( F `  y )  e.  x
)
3532, 34sylib 190 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
( F `  y
)  e.  x )
3635ralrimiva 2795 . . . . . . . . . . . 12  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  A. y  e.  dom  F ( F `
 y )  e.  x )
37 ffnfv 5923 . . . . . . . . . . . 12  |-  ( F : dom  F --> x  <->  ( F  Fn  dom  F  /\  A. y  e.  dom  F ( F `  y )  e.  x ) )
3823, 36, 37sylanbrc 647 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  F : dom  F --> x )
399, 22mp1i 12 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  F  Fn  dom  F )
4017ad2antrr 708 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  Smo  F )
41 smogt 6658 . . . . . . . . . . . . . . . 16  |-  ( ( F  Fn  dom  F  /\  Smo  F  /\  y  e.  dom  F )  -> 
y  C_  ( F `  y ) )
4239, 40, 28, 41syl3anc 1185 . . . . . . . . . . . . . . 15  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
y  C_  ( F `  y ) )
43 ordelon 4634 . . . . . . . . . . . . . . . . 17  |-  ( ( Ord  dom  F  /\  y  e.  dom  F )  ->  y  e.  On )
448, 28, 43sylancr 646 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
y  e.  On )
45 simpll 732 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  A  C_  On )
4645, 27sseldd 3335 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  x  e.  On )
47 ontr2 4657 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  On  /\  x  e.  On )  ->  ( ( y  C_  ( F `  y )  /\  ( F `  y )  e.  x
)  ->  y  e.  x ) )
4844, 46, 47syl2anc 644 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 662 . . . . . . . . . . . . . 14  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
y  e.  x )
5049ex 425 . . . . . . . . . . . . 13  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  (
y  e.  dom  F  ->  y  e.  x ) )
5150ssrdv 3340 . . . . . . . . . . . 12  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  dom  F 
C_  x )
5219, 51ssexd 4379 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  dom  F  e.  _V )
53 fex2 5632 . . . . . . . . . . 11  |-  ( ( F : dom  F --> x  /\  dom  F  e. 
_V  /\  x  e.  A )  ->  F  e.  _V )
5438, 52, 19, 53syl3anc 1185 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  F  e.  _V )
555ordtype2 7532 . . . . . . . . . 10  |-  ( (  _E  We  A  /\  _E Se  A  /\  F  e. 
_V )  ->  F  Isom  _E  ,  _E  ( dom  F ,  A ) )
5620, 21, 54, 55syl3anc 1185 . . . . . . . . 9  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  F  Isom  _E  ,  _E  ( dom  F ,  A ) )
57 isof1o 6074 . . . . . . . . 9  |-  ( F 
Isom  _E  ,  _E  ( dom  F ,  A
)  ->  F : dom  F -1-1-onto-> A )
58 f1ofo 5710 . . . . . . . . 9  |-  ( F : dom  F -1-1-onto-> A  ->  F : dom  F -onto-> A
)
59 forn 5685 . . . . . . . . 9  |-  ( F : dom  F -onto-> A  ->  ran  F  =  A )
6056, 57, 58, 594syl 20 . . . . . . . 8  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  ran  F  =  A )
6119, 60eleqtrrd 2519 . . . . . . 7  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  x  e.  ran  F )
6261expr 600 . . . . . 6  |-  ( ( A  C_  On  /\  x  e.  A )  ->  ( -.  x  e.  ran  F  ->  x  e.  ran  F ) )
6362pm2.18d 106 . . . . 5  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  ran  F )
6463ex 425 . . . 4  |-  ( A 
C_  On  ->  ( x  e.  A  ->  x  e.  ran  F ) )
6564ssrdv 3340 . . 3  |-  ( A 
C_  On  ->  A  C_  ran  F )
6618, 65eqssd 3351 . 2  |-  ( A 
C_  On  ->  ran  F  =  A )
6717, 66jca 520 1  |-  ( A 
C_  On  ->  ( Smo 
F  /\  ran  F  =  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1727   A.wral 2711   _Vcvv 2962    C_ wss 3306   class class class wbr 4237    _E cep 4521   Se wse 4568    We wwe 4569   Ord word 4609   Oncon0 4610   dom cdm 4907   ran crn 4908    Fn wfn 5478   -->wf 5479   -onto->wfo 5481   -1-1-onto->wf1o 5482   ` cfv 5483    Isom wiso 5484   Smo wsmo 6636  OrdIsocoi 7507
This theorem is referenced by:  oiid  7539  hsmexlem1  8337  hsmexlem2  8338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-riota 6578  df-smo 6637  df-recs 6662  df-oi 7508
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