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Theorem cantnflt2 7628
Description: An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
cantnflt2.4  |-  ( ph  ->  F  e.  S )
cantnflt2.5  |-  ( ph  -> 
(/)  e.  A )
cantnflt2.6  |-  ( ph  ->  C  e.  On )
cantnflt2.7  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  C
)
Assertion
Ref Expression
cantnflt2  |-  ( ph  ->  ( ( A CNF  B
) `  F )  e.  ( A  ^o  C
) )

Proof of Theorem cantnflt2
Dummy variables  k 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfs.1 . . 3  |-  S  =  dom  ( A CNF  B
)
2 cantnfs.2 . . 3  |-  ( ph  ->  A  e.  On )
3 cantnfs.3 . . 3  |-  ( ph  ->  B  e.  On )
4 eqid 2436 . . 3  |- OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
5 cantnflt2.4 . . 3  |-  ( ph  ->  F  e.  S )
6 eqid 2436 . . 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, 6cantnfval 7623 . 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 cantnflt2.5 . . 3  |-  ( ph  -> 
(/)  e.  A )
9 cantnflt2.6 . . . . 5  |-  ( ph  ->  C  e.  On )
10 cantnflt2.7 . . . . 5  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  C
)
119, 10ssexd 4350 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  _V )
124oion 7505 . . . 4  |-  ( ( `' F " ( _V 
\  1o ) )  e.  _V  ->  dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) )  e.  On )
13 sucidg 4659 . . . 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 19 . . 3  |-  ( ph  ->  dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) )  e. 
suc  dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) )
151, 2, 3, 4, 5cantnfcl 7622 . . . . . . . 8  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) )  e. 
om ) )
1615simpld 446 . . . . . . 7  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
174oiiso 7506 . . . . . . 7  |-  ( ( ( `' 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 643 . . . . . 6  |-  ( ph  -> OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) )  Isom  _E  ,  _E  ( dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) ,  ( `' F "
( _V  \  1o ) ) ) )
19 isof1o 6045 . . . . . 6  |-  (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 ) ) )
2018, 19syl 16 . . . . 5  |-  ( ph  -> OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) : dom OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) -1-1-onto-> ( `' F " ( _V 
\  1o ) ) )
21 f1ofo 5681 . . . . 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 ) ) )
22 foima 5658 . . . . 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 ) ) )
2320, 21, 223syl 19 . . . 4  |-  ( ph  ->  (OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) " dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) )  =  ( `' F " ( _V 
\  1o ) ) )
2423, 10eqsstrd 3382 . . 3  |-  ( ph  ->  (OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) " dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) )  C_  C
)
251, 2, 3, 4, 5, 6, 8, 14, 9, 24cantnflt 7627 . 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 ) )
267, 25eqeltrd 2510 1  |-  ( ph  ->  ( ( A CNF  B
) `  F )  e.  ( A  ^o  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317    C_ wss 3320   (/)c0 3628    _E cep 4492    We wwe 4540   Oncon0 4581   suc csuc 4583   omcom 4845   `'ccnv 4877   dom cdm 4878   "cima 4881   -onto->wfo 5452   -1-1-onto->wf1o 5453   ` cfv 5454    Isom wiso 5455  (class class class)co 6081    e. cmpt2 6083  seq𝜔cseqom 6704   1oc1o 6717    +o coa 6721    .o comu 6722    ^o coe 6723  OrdIsocoi 7478   CNF ccnf 7616
This theorem is referenced by:  cantnff  7629  cantnflem1d  7644  cnfcom3lem  7660
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-seqom 6705  df-1o 6724  df-2o 6725  df-oadd 6728  df-omul 6729  df-oexp 6730  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-cnf 7617
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