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Theorem cantnfclOLD 8068
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 8038 as of 28-Jun-2019. (New usage is discouraged.) (Proof modification 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 5298 . . . . 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 8067 . . . . . . . 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 457 . . . . . 6  |-  ( ph  ->  F : B --> A )
9 fdm 5674 . . . . . 6  |-  ( F : B --> A  ->  dom  F  =  B )
108, 9syl 17 . . . . 5  |-  ( ph  ->  dom  F  =  B )
111, 10syl5sseq 3489 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  B
)
12 onss 6564 . . . . 5  |-  ( B  e.  On  ->  B  C_  On )
135, 12syl 17 . . . 4  |-  ( ph  ->  B  C_  On )
1411, 13sstrd 3451 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  On )
15 epweon 6557 . . 3  |-  _E  We  On
16 wess 4809 . . 3  |-  ( ( `' F " ( _V 
\  1o ) ) 
C_  On  ->  (  _E  We  On  ->  _E  We  ( `' F "
( _V  \  1o ) ) ) )
1714, 15, 16mpisyl 19 . 2  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
18 cnvexg 6684 . . . . 5  |-  ( F  e.  S  ->  `' F  e.  _V )
19 imaexg 6675 . . . . 5  |-  ( `' F  e.  _V  ->  ( `' F " ( _V 
\  1o ) )  e.  _V )
20 cantnfvalOLD.3 . . . . . 6  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
2120oion 7915 . . . . 5  |-  ( ( `' F " ( _V 
\  1o ) )  e.  _V  ->  dom  G  e.  On )
222, 18, 19, 214syl 21 . . . 4  |-  ( ph  ->  dom  G  e.  On )
237simprd 461 . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  Fin )
2420oien 7917 . . . . . 6  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  Fin  /\  _E  We  ( `' F " ( _V 
\  1o ) ) )  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
2523, 17, 24syl2anc 659 . . . . 5  |-  ( ph  ->  dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )
26 enfii 7692 . . . . 5  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  Fin  /\ 
dom  G  ~~  ( `' F " ( _V 
\  1o ) ) )  ->  dom  G  e. 
Fin )
2723, 25, 26syl2anc 659 . . . 4  |-  ( ph  ->  dom  G  e.  Fin )
2822, 27elind 3626 . . 3  |-  ( ph  ->  dom  G  e.  ( On  i^i  Fin )
)
29 onfin2 7667 . . 3  |-  om  =  ( On  i^i  Fin )
3028, 29syl6eleqr 2501 . 2  |-  ( ph  ->  dom  G  e.  om )
3117, 30jca 530 1  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom  G  e. 
om ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058    \ cdif 3410    i^i cin 3412    C_ wss 3413   class class class wbr 4394    _E cep 4731    We wwe 4780   Oncon0 4821   `'ccnv 4941   dom cdm 4942   "cima 4945   -->wf 5521  (class class class)co 6234   omcom 6638   1oc1o 7080    ~~ cen 7471   Fincfn 7474  OrdIsocoi 7888   CNF ccnf 8030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-supp 6857  df-recs 6999  df-rdg 7033  df-seqom 7070  df-1o 7087  df-er 7268  df-map 7379  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-fsupp 7784  df-oi 7889  df-cnf 8031
This theorem is referenced by:  cantnfval2OLD  8070  cantnfleOLD  8072  cantnfltOLD  8073  cantnflt2OLD  8074  cantnfp1lem2OLD  8076  cantnfp1lem3OLD  8077  cantnflem1bOLD  8080  cantnflem1dOLD  8082  cantnflem1OLD  8083  cnfcomlemOLD  8103  cnfcomOLD  8104  cnfcom2lemOLD  8105  cnfcom3lemOLD  8107
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