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Theorem cantnfclOLD 8006
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.) Obsolete version of cantnfcl 7976 as of 28-Jun-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
cantnfsOLD.1  |-  S  =  dom  ( A CNF  B
)
cantnfsOLD.2  |-  ( ph  ->  A  e.  On )
cantnfsOLD.3  |-  ( ph  ->  B  e.  On )
cantnfvalOLD.3  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
cantnfvalOLD.4  |-  ( ph  ->  F  e.  S )
Assertion
Ref Expression
cantnfclOLD  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  G  e. 
om ) )

Proof of Theorem cantnfclOLD
StepHypRef Expression
1 cnvimass 5287 . . . . 5  |-  ( `' F " ( _V 
\  1o ) ) 
C_  dom  F
2 cantnfvalOLD.4 . . . . . . . 8  |-  ( ph  ->  F  e.  S )
3 cantnfsOLD.1 . . . . . . . . 9  |-  S  =  dom  ( A CNF  B
)
4 cantnfsOLD.2 . . . . . . . . 9  |-  ( ph  ->  A  e.  On )
5 cantnfsOLD.3 . . . . . . . . 9  |-  ( ph  ->  B  e.  On )
63, 4, 5cantnfsOLD 8005 . . . . . . . 8  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )
72, 6mpbid 210 . . . . . . 7  |-  ( ph  ->  ( F : B --> A  /\  ( `' F " ( _V  \  1o ) )  e.  Fin ) )
87simpld 459 . . . . . 6  |-  ( ph  ->  F : B --> A )
9 fdm 5661 . . . . . 6  |-  ( F : B --> A  ->  dom  F  =  B )
108, 9syl 16 . . . . 5  |-  ( ph  ->  dom  F  =  B )
111, 10syl5sseq 3502 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  B
)
12 onss 6502 . . . . 5  |-  ( B  e.  On  ->  B  C_  On )
135, 12syl 16 . . . 4  |-  ( ph  ->  B  C_  On )
1411, 13sstrd 3464 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  On )
15 epweon 6495 . . 3  |-  _E  We  On
16 wess 4805 . . 3  |-  ( ( `' F " ( _V 
\  1o ) ) 
C_  On  ->  (  _E  We  On  ->  _E  We  ( `' F "
( _V  \  1o ) ) ) )
1714, 15, 16mpisyl 18 . 2  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
18 cnvexg 6624 . . . . 5  |-  ( F  e.  S  ->  `' F  e.  _V )
19 imaexg 6615 . . . . 5  |-  ( `' F  e.  _V  ->  ( `' F " ( _V 
\  1o ) )  e.  _V )
20 cantnfvalOLD.3 . . . . . 6  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
2120oion 7851 . . . . 5  |-  ( ( `' F " ( _V 
\  1o ) )  e.  _V  ->  dom  G  e.  On )
222, 18, 19, 214syl 21 . . . 4  |-  ( ph  ->  dom  G  e.  On )
237simprd 463 . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  Fin )
2420oien 7853 . . . . . 6  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  Fin  /\  _E  We  ( `' F " ( _V 
\  1o ) ) )  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
2523, 17, 24syl2anc 661 . . . . 5  |-  ( ph  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
26 enfii 7631 . . . . 5  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  Fin  /\ 
dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )  ->  dom  G  e. 
Fin )
2723, 25, 26syl2anc 661 . . . 4  |-  ( ph  ->  dom  G  e.  Fin )
2822, 27elind 3638 . . 3  |-  ( ph  ->  dom  G  e.  ( On  i^i  Fin )
)
29 onfin2 7603 . . 3  |-  om  =  ( On  i^i  Fin )
3028, 29syl6eleqr 2550 . 2  |-  ( ph  ->  dom  G  e.  om )
3117, 30jca 532 1  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  G  e. 
om ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3068    \ cdif 3423    i^i cin 3425    C_ wss 3426   class class class wbr 4390    _E cep 4728    We wwe 4776   Oncon0 4817   `'ccnv 4937   dom cdm 4938   "cima 4941   -->wf 5512  (class class class)co 6190   omcom 6576   1oc1o 7013    ~~ cen 7407   Fincfn 7410  OrdIsocoi 7824   CNF ccnf 7968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-seqom 7003  df-1o 7020  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-oi 7825  df-cnf 7969
This theorem is referenced by:  cantnfval2OLD  8008  cantnfleOLD  8010  cantnfltOLD  8011  cantnflt2OLD  8012  cantnfp1lem2OLD  8014  cantnfp1lem3OLD  8015  cantnflem1bOLD  8018  cantnflem1dOLD  8020  cantnflem1OLD  8021  cnfcomlemOLD  8041  cnfcomOLD  8042  cnfcom2lemOLD  8043  cnfcom3lemOLD  8045
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