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Theorem cantnflt2 7374
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 2283 . . 3  |- OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
5 cantnflt2.4 . . 3  |-  ( ph  ->  F  e.  S )
6 eqid 2283 . . 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 7369 . 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 cnvexg 5208 . . . . 5  |-  ( F  e.  S  ->  `' F  e.  _V )
10 imaexg 5026 . . . . 5  |-  ( `' F  e.  _V  ->  ( `' F " ( _V 
\  1o ) )  e.  _V )
115, 9, 103syl 18 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  _V )
124oion 7251 . . . 4  |-  ( ( `' F " ( _V 
\  1o ) )  e.  _V  ->  dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) )  e.  On )
13 sucidg 4470 . . . 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 18 . . 3  |-  ( ph  ->  dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) )  e. 
suc  dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) )
15 cantnflt2.6 . . 3  |-  ( ph  ->  C  e.  On )
16 cantnflt2.7 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  C
)
17 ssexg 4160 . . . . . . . 8  |-  ( ( ( `' F "
( _V  \  1o ) )  C_  C  /\  C  e.  On )  ->  ( `' F " ( _V  \  1o ) )  e.  _V )
1816, 15, 17syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  _V )
191, 2, 3, 4, 5cantnfcl 7368 . . . . . . . 8  |-  ( ph  ->  (  _E  We  ( `' F " ( _V 
\  1o ) )  /\  dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) )  e. 
om ) )
2019simpld 445 . . . . . . 7  |-  ( ph  ->  _E  We  ( `' F " ( _V 
\  1o ) ) )
214oiiso 7252 . . . . . . 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 ) ) ) )
2218, 20, 21syl2anc 642 . . . . . 6  |-  ( ph  -> OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) )  Isom  _E  ,  _E  ( dom OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) ,  ( `' F "
( _V  \  1o ) ) ) )
23 isof1o 5822 . . . . . 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 ) ) )
2422, 23syl 15 . . . . 5  |-  ( ph  -> OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) : dom OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) -1-1-onto-> ( `' F " ( _V 
\  1o ) ) )
25 f1ofo 5479 . . . . 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 ) ) )
26 foima 5456 . . . . 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 ) ) )
2724, 25, 263syl 18 . . . 4  |-  ( ph  ->  (OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) " dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) )  =  ( `' F " ( _V 
\  1o ) ) )
2827, 16eqsstrd 3212 . . 3  |-  ( ph  ->  (OrdIso (  _E  , 
( `' F "
( _V  \  1o ) ) ) " dom OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) )  C_  C
)
291, 2, 3, 4, 5, 6, 8, 14, 15, 28cantnflt 7373 . 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 ) )
307, 29eqeltrd 2357 1  |-  ( ph  ->  ( ( A CNF  B
) `  F )  e.  ( A  ^o  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455    _E cep 4303    We wwe 4351   Oncon0 4392   suc csuc 4394   omcom 4656   `'ccnv 4688   dom cdm 4689   "cima 4692   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256  (class class class)co 5858    e. cmpt2 5860  seq𝜔cseqom 6459   1oc1o 6472    +o coa 6476    .o comu 6477    ^o coe 6478  OrdIsocoi 7224   CNF ccnf 7362
This theorem is referenced by:  cantnff  7375  cantnflem1d  7390  cnfcom3lem  7406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-oexp 6485  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-cnf 7363
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