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Theorem cantnfcl 7624
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 5226 . . . . 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 7623 . . . . . . . 8  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )
72, 6mpbid 203 . . . . . . 7  |-  ( ph  ->  ( F : B --> A  /\  ( `' F " ( _V  \  1o ) )  e.  Fin ) )
87simpld 447 . . . . . 6  |-  ( ph  ->  F : B --> A )
9 fdm 5597 . . . . . 6  |-  ( F : B --> A  ->  dom  F  =  B )
108, 9syl 16 . . . . 5  |-  ( ph  ->  dom  F  =  B )
111, 10syl5sseq 3398 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  B
)
12 onss 4773 . . . . 5  |-  ( B  e.  On  ->  B  C_  On )
135, 12syl 16 . . . 4  |-  ( ph  ->  B  C_  On )
1411, 13sstrd 3360 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  On )
15 epweon 4766 . . 3  |-  _E  We  On
16 wess 4571 . . 3  |-  ( ( `' F " ( _V 
\  1o ) ) 
C_  On  ->  (  _E  We  On  ->  _E  We  ( `' F "
( _V  \  1o ) ) ) )
1714, 15, 16ee10 1386 . 2  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
18 cnvexg 5407 . . . . 5  |-  ( F  e.  S  ->  `' F  e.  _V )
19 imaexg 5219 . . . . 5  |-  ( `' F  e.  _V  ->  ( `' F " ( _V 
\  1o ) )  e.  _V )
20 cantnfval.3 . . . . . 6  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
2120oion 7507 . . . . 5  |-  ( ( `' F " ( _V 
\  1o ) )  e.  _V  ->  dom  G  e.  On )
222, 18, 19, 214syl 20 . . . 4  |-  ( ph  ->  dom  G  e.  On )
237simprd 451 . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  Fin )
2420oien 7509 . . . . . 6  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  Fin  /\  _E  We  ( `' F " ( _V 
\  1o ) ) )  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
2523, 17, 24syl2anc 644 . . . . 5  |-  ( ph  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
26 enfii 7328 . . . . 5  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  Fin  /\ 
dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )  ->  dom  G  e. 
Fin )
2723, 25, 26syl2anc 644 . . . 4  |-  ( ph  ->  dom  G  e.  Fin )
28 elin 3532 . . . 4  |-  ( dom 
G  e.  ( On 
i^i  Fin )  <->  ( dom  G  e.  On  /\  dom  G  e.  Fin ) )
2922, 27, 28sylanbrc 647 . . 3  |-  ( ph  ->  dom  G  e.  ( On  i^i  Fin )
)
30 onfin2 7300 . . 3  |-  om  =  ( On  i^i  Fin )
3129, 30syl6eleqr 2529 . 2  |-  ( ph  ->  dom  G  e.  om )
3217, 31jca 520 1  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  G  e. 
om ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319    i^i cin 3321    C_ wss 3322   class class class wbr 4214    _E cep 4494    We wwe 4542   Oncon0 4583   omcom 4847   `'ccnv 4879   dom cdm 4880   "cima 4883   -->wf 5452  (class class class)co 6083   1oc1o 6719    ~~ cen 7108   Fincfn 7111  OrdIsocoi 7480   CNF ccnf 7618
This theorem is referenced by:  cantnfval2  7626  cantnfle  7628  cantnflt  7629  cantnflt2  7630  cantnff  7631  cantnfp1lem2  7637  cantnfp1lem3  7638  cantnflem1b  7644  cantnflem1d  7646  cantnflem1  7647  cnfcomlem  7658  cnfcom  7659  cnfcom2lem  7660  cnfcom3lem  7662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-recs 6635  df-rdg 6670  df-seqom 6707  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-oi 7481  df-cnf 7619
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