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Theorem oismo 7757
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 4406 (the second statement is trivial under ax-rep 4406). (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 6398 . . . . . 6  |-  _E  We  On
2 wess 4710 . . . . . 6  |-  ( A 
C_  On  ->  (  _E  We  On  ->  _E  We  A ) )
31, 2mpi 17 . . . . 5  |-  ( A 
C_  On  ->  _E  We  A )
4 epse 4706 . . . . 5  |-  _E Se  A
5 oismo.1 . . . . . 6  |-  F  = OrdIso
(  _E  ,  A
)
65oiiso2 7748 . . . . 5  |-  ( (  _E  We  A  /\  _E Se  A )  ->  F  Isom  _E  ,  _E  ( dom  F ,  ran  F
) )
73, 4, 6sylancl 662 . . . 4  |-  ( A 
C_  On  ->  F  Isom  _E  ,  _E  ( dom 
F ,  ran  F
) )
85oicl 7746 . . . . 5  |-  Ord  dom  F
95oif 7747 . . . . . . 7  |-  F : dom  F --> A
10 frn 5568 . . . . . . 7  |-  ( F : dom  F --> A  ->  ran  F  C_  A )
119, 10ax-mp 5 . . . . . 6  |-  ran  F  C_  A
12 id 22 . . . . . 6  |-  ( A 
C_  On  ->  A  C_  On )
1311, 12syl5ss 3370 . . . . 5  |-  ( A 
C_  On  ->  ran  F  C_  On )
14 smoiso2 6833 . . . . 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 663 . . . 4  |-  ( A 
C_  On  ->  ( ( F : dom  F -onto-> ran  F  /\  Smo  F
)  <->  F  Isom  _E  ,  _E  ( dom  F ,  ran  F ) ) )
167, 15mpbird 232 . . 3  |-  ( A 
C_  On  ->  ( F : dom  F -onto-> ran  F  /\  Smo  F ) )
1716simprd 463 . 2  |-  ( A 
C_  On  ->  Smo  F
)
1811a1i 11 . . 3  |-  ( A 
C_  On  ->  ran  F  C_  A )
19 simprl 755 . . . . . . . 8  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  x  e.  A )
203adantr 465 . . . . . . . . . 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 5562 . . . . . . . . . . . . 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 760 . . . . . . . . . . . . . . 15  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  -.  x  e.  ran  F )
253ad2antrr 725 . . . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  x  e.  A )
28 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
y  e.  dom  F
)
295oiiniseg 7750 . . . . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
( ( F `  y )  _E  x  \/  x  e.  ran  F ) )
3130ord 377 . . . . . . . . . . . . . . 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 125 . . . . . . . . . . . . . 14  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
( F `  y
)  _E  x )
33 vex 2978 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
3433epelc 4637 . . . . . . . . . . . . . 14  |-  ( ( F `  y )  _E  x  <->  ( F `  y )  e.  x
)
3532, 34sylib 196 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
( F `  y
)  e.  x )
3635ralrimiva 2802 . . . . . . . . . . . 12  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  A. y  e.  dom  F ( F `
 y )  e.  x )
37 ffnfv 5872 . . . . . . . . . . . 12  |-  ( F : dom  F --> x  <->  ( F  Fn  dom  F  /\  A. y  e.  dom  F ( F `  y )  e.  x ) )
3823, 36, 37sylanbrc 664 . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  Smo  F )
41 smogt 6831 . . . . . . . . . . . . . . . 16  |-  ( ( F  Fn  dom  F  /\  Smo  F  /\  y  e.  dom  F )  -> 
y  C_  ( F `  y ) )
4239, 40, 28, 41syl3anc 1218 . . . . . . . . . . . . . . 15  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
y  C_  ( F `  y ) )
43 ordelon 4746 . . . . . . . . . . . . . . . . 17  |-  ( ( Ord  dom  F  /\  y  e.  dom  F )  ->  y  e.  On )
448, 28, 43sylancr 663 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
y  e.  On )
45 simpll 753 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  A  C_  On )
4645, 27sseldd 3360 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  ->  x  e.  On )
47 ontr2 4769 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  On  /\  x  e.  On )  ->  ( ( y  C_  ( F `  y )  /\  ( F `  y )  e.  x
)  ->  y  e.  x ) )
4844, 46, 47syl2anc 661 . . . . . . . . . . . . . . 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 679 . . . . . . . . . . . . . 14  |-  ( ( ( A  C_  On  /\  ( x  e.  A  /\  -.  x  e.  ran  F ) )  /\  y  e.  dom  F )  -> 
y  e.  x )
5049ex 434 . . . . . . . . . . . . 13  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  (
y  e.  dom  F  ->  y  e.  x ) )
5150ssrdv 3365 . . . . . . . . . . . 12  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  dom  F 
C_  x )
5219, 51ssexd 4442 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  dom  F  e.  _V )
53 fex2 6535 . . . . . . . . . . 11  |-  ( ( F : dom  F --> x  /\  dom  F  e. 
_V  /\  x  e.  A )  ->  F  e.  _V )
5438, 52, 19, 53syl3anc 1218 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  F  e.  _V )
555ordtype2 7751 . . . . . . . . . 10  |-  ( (  _E  We  A  /\  _E Se  A  /\  F  e. 
_V )  ->  F  Isom  _E  ,  _E  ( dom  F ,  A ) )
5620, 21, 54, 55syl3anc 1218 . . . . . . . . 9  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  F  Isom  _E  ,  _E  ( dom  F ,  A ) )
57 isof1o 6019 . . . . . . . . 9  |-  ( F 
Isom  _E  ,  _E  ( dom  F ,  A
)  ->  F : dom  F -1-1-onto-> A )
58 f1ofo 5651 . . . . . . . . 9  |-  ( F : dom  F -1-1-onto-> A  ->  F : dom  F -onto-> A
)
59 forn 5626 . . . . . . . . 9  |-  ( F : dom  F -onto-> A  ->  ran  F  =  A )
6056, 57, 58, 594syl 21 . . . . . . . 8  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  ran  F  =  A )
6119, 60eleqtrrd 2520 . . . . . . 7  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  -.  x  e.  ran  F ) )  ->  x  e.  ran  F )
6261expr 615 . . . . . 6  |-  ( ( A  C_  On  /\  x  e.  A )  ->  ( -.  x  e.  ran  F  ->  x  e.  ran  F ) )
6362pm2.18d 111 . . . . 5  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  ran  F )
6463ex 434 . . . 4  |-  ( A 
C_  On  ->  ( x  e.  A  ->  x  e.  ran  F ) )
6564ssrdv 3365 . . 3  |-  ( A 
C_  On  ->  A  C_  ran  F )
6618, 65eqssd 3376 . 2  |-  ( A 
C_  On  ->  ran  F  =  A )
6717, 66jca 532 1  |-  ( A 
C_  On  ->  ( Smo 
F  /\  ran  F  =  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2718   _Vcvv 2975    C_ wss 3331   class class class wbr 4295    _E cep 4633   Se wse 4680    We wwe 4681   Ord word 4721   Oncon0 4722   dom cdm 4843   ran crn 4844    Fn wfn 5416   -->wf 5417   -onto->wfo 5419   -1-1-onto->wf1o 5420   ` cfv 5421    Isom wiso 5422   Smo wsmo 6809  OrdIsocoi 7726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-smo 6810  df-recs 6835  df-oi 7727
This theorem is referenced by:  oiid  7758  hsmexlem1  8598  hsmexlem2  8599
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