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Theorem cantnflt2OLD 8139
Description: An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.) Obsolete version of cantnflt2 8109 as of 29-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 )
cantnflt2OLD.4  |-  ( ph  ->  F  e.  S )
cantnflt2OLD.5  |-  ( ph  -> 
(/)  e.  A )
cantnflt2OLD.6  |-  ( ph  ->  C  e.  On )
cantnflt2OLD.7  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  C
)
Assertion
Ref Expression
cantnflt2OLD  |-  ( ph  ->  ( ( A CNF  B
) `  F )  e.  ( A  ^o  C
) )

Proof of Theorem cantnflt2OLD
Dummy variables  k 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfsOLD.1 . . 3  |-  S  =  dom  ( A CNF  B
)
2 cantnfsOLD.2 . . 3  |-  ( ph  ->  A  e.  On )
3 cantnfsOLD.3 . . 3  |-  ( ph  ->  B  e.  On )
4 eqid 2457 . . 3  |- OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
5 cantnflt2OLD.4 . . 3  |-  ( ph  ->  F  e.  S )
6 eqid 2457 . . 3  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) `  k ) )  .o  ( F `  (OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )  = seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) `  k ) )  .o  ( F `  (OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )
71, 2, 3, 4, 5, 6cantnfvalOLD 8134 . 2  |-  ( ph  ->  ( ( A CNF  B
) `  F )  =  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) `  k ) )  .o  ( F `  (OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) ) )
8 cantnflt2OLD.5 . . 3  |-  ( ph  -> 
(/)  e.  A )
9 cantnflt2OLD.6 . . . . 5  |-  ( ph  ->  C  e.  On )
10 cantnflt2OLD.7 . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  C
)
119, 10ssexd 4603 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  _V )
124oion 7979 . . . 4  |-  ( ( `' F " ( _V 
\  1o ) )  e.  _V  ->  dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) )  e.  On )
13 sucidg 4965 . . . 4  |-  ( dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) )  e.  On  ->  dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) )  e.  suc  dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) )
1411, 12, 133syl 20 . . 3  |-  ( ph  ->  dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) )  e. 
suc  dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) )
151, 2, 3, 4, 5cantnfclOLD 8133 . . . . . . 7  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) )  e. 
om ) )
1615simpld 459 . . . . . 6  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
174oiiso 7980 . . . . . 6  |-  ( ( ( `' F "
( _V  \  1o ) )  e.  _V  /\  _E  We  ( `' F " ( _V 
\  1o ) ) )  -> OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) ,  ( `' F " ( _V 
\  1o ) ) ) )
1811, 16, 17syl2anc 661 . . . . 5  |-  ( ph  -> OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) )  Isom  _E  ,  _E  ( dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) ,  ( `' F "
( _V  \  1o ) ) ) )
19 isof1o 6222 . . . . 5  |-  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) ,  ( `' F " ( _V 
\  1o ) ) )  -> OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) : dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) -1-1-onto-> ( `' F " ( _V 
\  1o ) ) )
20 f1ofo 5829 . . . . 5  |-  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) : dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) -1-1-onto-> ( `' F " ( _V 
\  1o ) )  -> OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) : dom OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) -onto-> ( `' F " ( _V 
\  1o ) ) )
21 foima 5806 . . . . 5  |-  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) : dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) -onto-> ( `' F " ( _V 
\  1o ) )  ->  (OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) " dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) )  =  ( `' F " ( _V 
\  1o ) ) )
2218, 19, 20, 214syl 21 . . . 4  |-  ( ph  ->  (OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) " dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) )  =  ( `' F " ( _V 
\  1o ) ) )
2322, 10eqsstrd 3533 . . 3  |-  ( ph  ->  (OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) " dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) )  C_  C
)
241, 2, 3, 4, 5, 6, 8, 14, 9, 23cantnfltOLD 8138 . 2  |-  ( ph  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) `  k ) )  .o  ( F `  (OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) )  e.  ( A  ^o  C ) )
257, 24eqeltrd 2545 1  |-  ( ph  ->  ( ( A CNF  B
) `  F )  e.  ( A  ^o  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   _Vcvv 3109    \ cdif 3468    C_ wss 3471   (/)c0 3793    _E cep 4798    We wwe 4846   Oncon0 4887   suc csuc 4889   `'ccnv 5007   dom cdm 5008   "cima 5011   -onto->wfo 5592   -1-1-onto->wf1o 5593   ` cfv 5594    Isom wiso 5595  (class class class)co 6296    |-> cmpt2 6298   omcom 6699  seq𝜔cseqom 7130   1oc1o 7141    +o coa 7145    .o comu 7146    ^o coe 7147  OrdIsocoi 7952   CNF ccnf 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-seqom 7131  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-oexp 7154  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-cnf 8096
This theorem is referenced by:  cantnflem1dOLD  8147  cnfcom3lemOLD  8172
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