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Theorem cantnfcl 8082
Description: Basic properties of the order isomorphism  G used later. The support of an  F  e.  S is a finite subset of  A, so it is well-ordered by  _E and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
cantnfcl.g  |-  G  = OrdIso
(  _E  ,  ( F supp  (/) ) )
cantnfcl.f  |-  ( ph  ->  F  e.  S )
Assertion
Ref Expression
cantnfcl  |-  ( ph  ->  (  _E  We  ( F supp 
(/) )  /\  dom  G  e.  om ) )

Proof of Theorem cantnfcl
StepHypRef Expression
1 suppssdm 6911 . . . . 5  |-  ( F supp  (/) )  C_  dom  F
2 cantnfcl.f . . . . . . . 8  |-  ( ph  ->  F  e.  S )
3 cantnfs.s . . . . . . . . 9  |-  S  =  dom  ( A CNF  B
)
4 cantnfs.a . . . . . . . . 9  |-  ( ph  ->  A  e.  On )
5 cantnfs.b . . . . . . . . 9  |-  ( ph  ->  B  e.  On )
63, 4, 5cantnfs 8081 . . . . . . . 8  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  F finSupp 
(/) ) ) )
72, 6mpbid 210 . . . . . . 7  |-  ( ph  ->  ( F : B --> A  /\  F finSupp  (/) ) )
87simpld 459 . . . . . 6  |-  ( ph  ->  F : B --> A )
9 fdm 5733 . . . . . 6  |-  ( F : B --> A  ->  dom  F  =  B )
108, 9syl 16 . . . . 5  |-  ( ph  ->  dom  F  =  B )
111, 10syl5sseq 3552 . . . 4  |-  ( ph  ->  ( F supp  (/) )  C_  B )
12 onss 6604 . . . . 5  |-  ( B  e.  On  ->  B  C_  On )
135, 12syl 16 . . . 4  |-  ( ph  ->  B  C_  On )
1411, 13sstrd 3514 . . 3  |-  ( ph  ->  ( F supp  (/) )  C_  On )
15 epweon 6597 . . 3  |-  _E  We  On
16 wess 4866 . . 3  |-  ( ( F supp  (/) )  C_  On  ->  (  _E  We  On  ->  _E  We  ( F supp  (/) ) ) )
1714, 15, 16mpisyl 18 . 2  |-  ( ph  ->  _E  We  ( F supp  (/) ) )
18 ovex 6307 . . . . . 6  |-  ( F supp  (/) )  e.  _V
1918a1i 11 . . . . 5  |-  ( ph  ->  ( F supp  (/) )  e. 
_V )
20 cantnfcl.g . . . . . 6  |-  G  = OrdIso
(  _E  ,  ( F supp  (/) ) )
2120oion 7957 . . . . 5  |-  ( ( F supp  (/) )  e.  _V  ->  dom  G  e.  On )
2219, 21syl 16 . . . 4  |-  ( ph  ->  dom  G  e.  On )
237simprd 463 . . . . . 6  |-  ( ph  ->  F finSupp  (/) )
2423fsuppimpd 7832 . . . . 5  |-  ( ph  ->  ( F supp  (/) )  e. 
Fin )
2520oien 7959 . . . . . 6  |-  ( ( ( F supp  (/) )  e. 
_V  /\  _E  We  ( F supp  (/) ) )  ->  dom  G  ~~  ( F supp  (/) ) )
2619, 17, 25syl2anc 661 . . . . 5  |-  ( ph  ->  dom  G  ~~  ( F supp 
(/) ) )
27 enfii 7734 . . . . 5  |-  ( ( ( F supp  (/) )  e. 
Fin  /\  dom  G  ~~  ( F supp  (/) ) )  ->  dom  G  e.  Fin )
2824, 26, 27syl2anc 661 . . . 4  |-  ( ph  ->  dom  G  e.  Fin )
2922, 28elind 3688 . . 3  |-  ( ph  ->  dom  G  e.  ( On  i^i  Fin )
)
30 onfin2 7706 . . 3  |-  om  =  ( On  i^i  Fin )
3129, 30syl6eleqr 2566 . 2  |-  ( ph  ->  dom  G  e.  om )
3217, 31jca 532 1  |-  ( ph  ->  (  _E  We  ( F supp 
(/) )  /\  dom  G  e.  om ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785   class class class wbr 4447    _E cep 4789    We wwe 4837   Oncon0 4878   dom cdm 4999   -->wf 5582  (class class class)co 6282   omcom 6678   supp csupp 6898    ~~ cen 7510   Fincfn 7513   finSupp cfsupp 7825  OrdIsocoi 7930   CNF ccnf 8074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-supp 6899  df-recs 7039  df-rdg 7073  df-seqom 7110  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-oi 7931  df-cnf 8075
This theorem is referenced by:  cantnfval2  8084  cantnfle  8086  cantnflt  8087  cantnflt2  8088  cantnff  8089  cantnfp1lem2  8094  cantnfp1lem3  8095  cantnflem1b  8101  cantnflem1d  8103  cantnflem1  8104  cnfcomlem  8139  cnfcom  8140  cnfcom2lem  8141  cnfcom3lem  8143
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