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Theorem cantnfcl 7368
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 5033 . . . . 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 7367 . . . . . . . 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 5393 . . . . . 6  |-  ( F : B --> A  ->  dom  F  =  B )
108, 9syl 15 . . . . 5  |-  ( ph  ->  dom  F  =  B )
111, 10syl5sseq 3226 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  B
)
12 onss 4582 . . . . 5  |-  ( B  e.  On  ->  B  C_  On )
135, 12syl 15 . . . 4  |-  ( ph  ->  B  C_  On )
1411, 13sstrd 3189 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  On )
15 epweon 4575 . . 3  |-  _E  We  On
16 wess 4380 . . 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 5208 . . . . . 6  |-  ( F  e.  S  ->  `' F  e.  _V )
192, 18syl 15 . . . . 5  |-  ( ph  ->  `' F  e.  _V )
20 imaexg 5026 . . . . 5  |-  ( `' F  e.  _V  ->  ( `' F " ( _V 
\  1o ) )  e.  _V )
21 cantnfval.3 . . . . . 6  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
2221oion 7251 . . . . 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 7253 . . . . . 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 7080 . . . . 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 3358 . . . 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 7052 . . 3  |-  om  =  ( On  i^i  Fin )
3230, 31syl6eleqr 2374 . 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 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   class class class wbr 4023    _E cep 4303    We wwe 4351   Oncon0 4392   omcom 4656   `'ccnv 4688   dom cdm 4689   "cima 4692   -->wf 5251  (class class class)co 5858   1oc1o 6472    ~~ cen 6860   Fincfn 6863  OrdIsocoi 7224   CNF ccnf 7362
This theorem is referenced by:  cantnfval2  7370  cantnfle  7372  cantnflt  7373  cantnflt2  7374  cantnff  7375  cantnfp1lem2  7381  cantnfp1lem3  7382  cantnflem1b  7388  cantnflem1d  7390  cantnflem1  7391  cnfcomlem  7402  cnfcom  7403  cnfcom2lem  7404  cnfcom3lem  7406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-cnf 7363
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