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Theorem cantnfclOLD 8107
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 8077 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 5350 . . . . 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 8106 . . . . . . . 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 5728 . . . . . 6  |-  ( F : B --> A  ->  dom  F  =  B )
108, 9syl 16 . . . . 5  |-  ( ph  ->  dom  F  =  B )
111, 10syl5sseq 3547 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  B
)
12 onss 6599 . . . . 5  |-  ( B  e.  On  ->  B  C_  On )
135, 12syl 16 . . . 4  |-  ( ph  ->  B  C_  On )
1411, 13sstrd 3509 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  On )
15 epweon 6592 . . 3  |-  _E  We  On
16 wess 4861 . . 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 6722 . . . . 5  |-  ( F  e.  S  ->  `' F  e.  _V )
19 imaexg 6713 . . . . 5  |-  ( `' F  e.  _V  ->  ( `' F " ( _V 
\  1o ) )  e.  _V )
20 cantnfvalOLD.3 . . . . . 6  |-  G  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
2120oion 7952 . . . . 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 7954 . . . . . 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 7729 . . . . 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 3683 . . 3  |-  ( ph  ->  dom  G  e.  ( On  i^i  Fin )
)
29 onfin2 7701 . . 3  |-  om  =  ( On  i^i  Fin )
3028, 29syl6eleqr 2561 . 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 1374    e. wcel 1762   _Vcvv 3108    \ cdif 3468    i^i cin 3470    C_ wss 3471   class class class wbr 4442    _E cep 4784    We wwe 4832   Oncon0 4873   `'ccnv 4993   dom cdm 4994   "cima 4997   -->wf 5577  (class class class)co 6277   omcom 6673   1oc1o 7115    ~~ cen 7505   Fincfn 7508  OrdIsocoi 7925   CNF ccnf 8069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-seqom 7105  df-1o 7122  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-oi 7926  df-cnf 8070
This theorem is referenced by:  cantnfval2OLD  8109  cantnfleOLD  8111  cantnfltOLD  8112  cantnflt2OLD  8113  cantnfp1lem2OLD  8115  cantnfp1lem3OLD  8116  cantnflem1bOLD  8119  cantnflem1dOLD  8121  cantnflem1OLD  8122  cnfcomlemOLD  8142  cnfcomOLD  8143  cnfcom2lemOLD  8144  cnfcom3lemOLD  8146
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