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Theorem cantnfcl 7384
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.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
cantnfval.3  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
cantnfval.4  |-  ( ph  ->  F  e.  S )
Assertion
Ref Expression
cantnfcl  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  G  e. 
om ) )

Proof of Theorem cantnfcl
StepHypRef Expression
1 cnvimass 5049 . . . . 5  |-  ( `' F " ( _V 
\  1o ) ) 
C_  dom  F
2 cantnfval.4 . . . . . . . 8  |-  ( ph  ->  F  e.  S )
3 cantnfs.1 . . . . . . . . 9  |-  S  =  dom  ( A CNF  B
)
4 cantnfs.2 . . . . . . . . 9  |-  ( ph  ->  A  e.  On )
5 cantnfs.3 . . . . . . . . 9  |-  ( ph  ->  B  e.  On )
63, 4, 5cantnfs 7383 . . . . . . . 8  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )
72, 6mpbid 201 . . . . . . 7  |-  ( ph  ->  ( F : B --> A  /\  ( `' F " ( _V  \  1o ) )  e.  Fin ) )
87simpld 445 . . . . . 6  |-  ( ph  ->  F : B --> A )
9 fdm 5409 . . . . . 6  |-  ( F : B --> A  ->  dom  F  =  B )
108, 9syl 15 . . . . 5  |-  ( ph  ->  dom  F  =  B )
111, 10syl5sseq 3239 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  B
)
12 onss 4598 . . . . 5  |-  ( B  e.  On  ->  B  C_  On )
135, 12syl 15 . . . 4  |-  ( ph  ->  B  C_  On )
1411, 13sstrd 3202 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  On )
15 epweon 4591 . . 3  |-  _E  We  On
16 wess 4396 . . 3  |-  ( ( `' F " ( _V 
\  1o ) ) 
C_  On  ->  (  _E  We  On  ->  _E  We  ( `' F "
( _V  \  1o ) ) ) )
1714, 15, 16ee10 1366 . 2  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
18 cnvexg 5224 . . . . . 6  |-  ( F  e.  S  ->  `' F  e.  _V )
192, 18syl 15 . . . . 5  |-  ( ph  ->  `' F  e.  _V )
20 imaexg 5042 . . . . 5  |-  ( `' F  e.  _V  ->  ( `' F " ( _V 
\  1o ) )  e.  _V )
21 cantnfval.3 . . . . . 6  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
2221oion 7267 . . . . 5  |-  ( ( `' F " ( _V 
\  1o ) )  e.  _V  ->  dom  G  e.  On )
2319, 20, 223syl 18 . . . 4  |-  ( ph  ->  dom  G  e.  On )
247simprd 449 . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  Fin )
2521oien 7269 . . . . . 6  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  Fin  /\  _E  We  ( `' F " ( _V 
\  1o ) ) )  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
2624, 17, 25syl2anc 642 . . . . 5  |-  ( ph  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
27 enfii 7096 . . . . 5  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  Fin  /\ 
dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )  ->  dom  G  e. 
Fin )
2824, 26, 27syl2anc 642 . . . 4  |-  ( ph  ->  dom  G  e.  Fin )
29 elin 3371 . . . 4  |-  ( dom 
G  e.  ( On 
i^i  Fin )  <->  ( dom  G  e.  On  /\  dom  G  e.  Fin ) )
3023, 28, 29sylanbrc 645 . . 3  |-  ( ph  ->  dom  G  e.  ( On  i^i  Fin )
)
31 onfin2 7068 . . 3  |-  om  =  ( On  i^i  Fin )
3230, 31syl6eleqr 2387 . 2  |-  ( ph  ->  dom  G  e.  om )
3317, 32jca 518 1  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  G  e. 
om ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   class class class wbr 4039    _E cep 4319    We wwe 4367   Oncon0 4408   omcom 4672   `'ccnv 4704   dom cdm 4705   "cima 4708   -->wf 5267  (class class class)co 5874   1oc1o 6488    ~~ cen 6876   Fincfn 6879  OrdIsocoi 7240   CNF ccnf 7378
This theorem is referenced by:  cantnfval2  7386  cantnfle  7388  cantnflt  7389  cantnflt2  7390  cantnff  7391  cantnfp1lem2  7397  cantnfp1lem3  7398  cantnflem1b  7404  cantnflem1d  7406  cantnflem1  7407  cnfcomlem  7418  cnfcom  7419  cnfcom2lem  7420  cnfcom3lem  7422
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-rep 4147  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-om 4673  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-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-seqom 6476  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-cnf 7379
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