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Theorem cantnflt2 8124
Description: An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
cantnflt2.f  |-  ( ph  ->  F  e.  S )
cantnflt2.a  |-  ( ph  -> 
(/)  e.  A )
cantnflt2.c  |-  ( ph  ->  C  e.  On )
cantnflt2.s  |-  ( ph  ->  ( F supp  (/) )  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.s . . 3  |-  S  =  dom  ( A CNF  B
)
2 cantnfs.a . . 3  |-  ( ph  ->  A  e.  On )
3 cantnfs.b . . 3  |-  ( ph  ->  B  e.  On )
4 eqid 2402 . . 3  |- OrdIso (  _E  ,  ( F supp  (/) ) )  = OrdIso (  _E  , 
( F supp  (/) ) )
5 cantnflt2.f . . 3  |-  ( ph  ->  F  e.  S )
6 eqid 2402 . . 3  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( F supp  (/) ) ) `
 k ) )  .o  ( F `  (OrdIso (  _E  ,  ( F supp  (/) ) ) `  k ) ) )  +o  z ) ) ,  (/) )  = seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( F supp  (/) ) ) `  k ) )  .o  ( F `  (OrdIso (  _E  ,  ( F supp 
(/) ) ) `  k ) ) )  +o  z ) ) ,  (/) )
71, 2, 3, 4, 5, 6cantnfval 8119 . 2  |-  ( ph  ->  ( ( A CNF  B
) `  F )  =  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( F supp  (/) ) ) `
 k ) )  .o  ( F `  (OrdIso (  _E  ,  ( F supp  (/) ) ) `  k ) ) )  +o  z ) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( F supp  (/) ) ) ) )
8 cantnflt2.a . . 3  |-  ( ph  -> 
(/)  e.  A )
9 cantnflt2.c . . . . 5  |-  ( ph  ->  C  e.  On )
10 cantnflt2.s . . . . 5  |-  ( ph  ->  ( F supp  (/) )  C_  C )
119, 10ssexd 4541 . . . 4  |-  ( ph  ->  ( F supp  (/) )  e. 
_V )
124oion 7995 . . . 4  |-  ( ( F supp  (/) )  e.  _V  ->  dom OrdIso (  _E  , 
( F supp  (/) ) )  e.  On )
13 sucidg 5488 . . . 4  |-  ( dom OrdIso (  _E  ,  ( F supp 
(/) ) )  e.  On  ->  dom OrdIso (  _E  ,  ( F supp  (/) ) )  e.  suc  dom OrdIso (  _E  ,  ( F supp  (/) ) ) )
1411, 12, 133syl 18 . . 3  |-  ( ph  ->  dom OrdIso (  _E  , 
( F supp  (/) ) )  e.  suc  dom OrdIso (  _E  ,  ( F supp  (/) ) ) )
151, 2, 3, 4, 5cantnfcl 8118 . . . . . . 7  |-  ( ph  ->  (  _E  We  ( F supp 
(/) )  /\  dom OrdIso (  _E  ,  ( F supp  (/) ) )  e.  om ) )
1615simpld 457 . . . . . 6  |-  ( ph  ->  _E  We  ( F supp  (/) ) )
174oiiso 7996 . . . . . 6  |-  ( ( ( F supp  (/) )  e. 
_V  /\  _E  We  ( F supp  (/) ) )  -> OrdIso (  _E  ,  ( F supp  (/) ) )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  ( F supp 
(/) ) ) ,  ( F supp  (/) ) ) )
1811, 16, 17syl2anc 659 . . . . 5  |-  ( ph  -> OrdIso (  _E  ,  ( F supp  (/) ) )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  ( F supp 
(/) ) ) ,  ( F supp  (/) ) ) )
19 isof1o 6204 . . . . 5  |-  (OrdIso (  _E  ,  ( F supp  (/) ) ) 
Isom  _E  ,  _E  ( dom OrdIso (  _E  , 
( F supp  (/) ) ) ,  ( F supp  (/) ) )  -> OrdIso (  _E  ,  ( F supp  (/) ) ) : dom OrdIso (  _E  , 
( F supp  (/) ) ) -1-1-onto-> ( F supp  (/) ) )
20 f1ofo 5806 . . . . 5  |-  (OrdIso (  _E  ,  ( F supp  (/) ) ) : dom OrdIso (  _E  , 
( F supp  (/) ) ) -1-1-onto-> ( F supp  (/) )  -> OrdIso (  _E  ,  ( F supp  (/) ) ) : dom OrdIso (  _E  , 
( F supp  (/) ) )
-onto-> ( F supp  (/) ) )
21 foima 5783 . . . . 5  |-  (OrdIso (  _E  ,  ( F supp  (/) ) ) : dom OrdIso (  _E  , 
( F supp  (/) ) )
-onto-> ( F supp  (/) )  -> 
(OrdIso (  _E  , 
( F supp  (/) ) )
" dom OrdIso (  _E  , 
( F supp  (/) ) ) )  =  ( F supp  (/) ) )
2218, 19, 20, 214syl 19 . . . 4  |-  ( ph  ->  (OrdIso (  _E  , 
( F supp  (/) ) )
" dom OrdIso (  _E  , 
( F supp  (/) ) ) )  =  ( F supp  (/) ) )
2322, 10eqsstrd 3476 . . 3  |-  ( ph  ->  (OrdIso (  _E  , 
( F supp  (/) ) )
" dom OrdIso (  _E  , 
( F supp  (/) ) ) )  C_  C )
241, 2, 3, 4, 5, 6, 8, 14, 9, 23cantnflt 8123 . 2  |-  ( ph  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( F supp  (/) ) ) `
 k ) )  .o  ( F `  (OrdIso (  _E  ,  ( F supp  (/) ) ) `  k ) ) )  +o  z ) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( F supp  (/) ) ) )  e.  ( A  ^o  C
) )
257, 24eqeltrd 2490 1  |-  ( ph  ->  ( ( A CNF  B
) `  F )  e.  ( A  ^o  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   _Vcvv 3059    C_ wss 3414   (/)c0 3738    _E cep 4732    We wwe 4781   dom cdm 4823   "cima 4826   Oncon0 5410   suc csuc 5412   -onto->wfo 5567   -1-1-onto->wf1o 5568   ` cfv 5569    Isom wiso 5570  (class class class)co 6278    |-> cmpt2 6280   omcom 6683   supp csupp 6902  seq𝜔cseqom 7149    +o coa 7164    .o comu 7165    ^o coe 7166  OrdIsocoi 7968   CNF ccnf 8110
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 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
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 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-seqom 7150  df-1o 7167  df-2o 7168  df-oadd 7171  df-omul 7172  df-oexp 7173  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-oi 7969  df-cnf 8111
This theorem is referenced by:  cantnff  8125  cantnflem1d  8139  cnfcom3lem  8179
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