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Theorem cantnfcl 8103
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 6930 . . . . 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 8102 . . . . . . . 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 5741 . . . . . 6  |-  ( F : B --> A  ->  dom  F  =  B )
108, 9syl 16 . . . . 5  |-  ( ph  ->  dom  F  =  B )
111, 10syl5sseq 3547 . . . 4  |-  ( ph  ->  ( F supp  (/) )  C_  B )
12 onss 6625 . . . . 5  |-  ( B  e.  On  ->  B  C_  On )
135, 12syl 16 . . . 4  |-  ( ph  ->  B  C_  On )
1411, 13sstrd 3509 . . 3  |-  ( ph  ->  ( F supp  (/) )  C_  On )
15 epweon 6618 . . 3  |-  _E  We  On
16 wess 4875 . . 3  |-  ( ( F supp  (/) )  C_  On  ->  (  _E  We  On  ->  _E  We  ( F supp  (/) ) ) )
1714, 15, 16mpisyl 18 . 2  |-  ( ph  ->  _E  We  ( F supp  (/) ) )
18 ovex 6324 . . . . . 6  |-  ( F supp  (/) )  e.  _V
1918a1i 11 . . . . 5  |-  ( ph  ->  ( F supp  (/) )  e. 
_V )
20 cantnfcl.g . . . . . 6  |-  G  = OrdIso
(  _E  ,  ( F supp  (/) ) )
2120oion 7979 . . . . 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 7854 . . . . 5  |-  ( ph  ->  ( F supp  (/) )  e. 
Fin )
2520oien 7981 . . . . . 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 7756 . . . . 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 3684 . . 3  |-  ( ph  ->  dom  G  e.  ( On  i^i  Fin )
)
30 onfin2 7728 . . 3  |-  om  =  ( On  i^i  Fin )
3129, 30syl6eleqr 2556 . 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 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   class class class wbr 4456    _E cep 4798    We wwe 4846   Oncon0 4887   dom cdm 5008   -->wf 5590  (class class class)co 6296   omcom 6699   supp csupp 6917    ~~ cen 7532   Fincfn 7535   finSupp cfsupp 7847  OrdIsocoi 7952   CNF ccnf 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-supp 6918  df-recs 7060  df-rdg 7094  df-seqom 7131  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-cnf 8096
This theorem is referenced by:  cantnfval2  8105  cantnfle  8107  cantnflt  8108  cantnflt2  8109  cantnff  8110  cantnfp1lem2  8115  cantnfp1lem3  8116  cantnflem1b  8122  cantnflem1d  8124  cantnflem1  8125  cnfcomlem  8160  cnfcom  8161  cnfcom2lem  8162  cnfcom3lem  8164
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