MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnflt2 Structured version   Unicode version

Theorem cantnflt2 7984
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 2451 . . 3  |- OrdIso (  _E  ,  ( F supp  (/) ) )  = OrdIso (  _E  , 
( F supp  (/) ) )
5 cantnflt2.f . . 3  |-  ( ph  ->  F  e.  S )
6 eqid 2451 . . 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 7979 . 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 4539 . . . 4  |-  ( ph  ->  ( F supp  (/) )  e. 
_V )
124oion 7853 . . . 4  |-  ( ( F supp  (/) )  e.  _V  ->  dom OrdIso (  _E  , 
( F supp  (/) ) )  e.  On )
13 sucidg 4897 . . . 4  |-  ( dom OrdIso (  _E  ,  ( F supp 
(/) ) )  e.  On  ->  dom OrdIso (  _E  ,  ( F supp  (/) ) )  e.  suc  dom OrdIso (  _E  ,  ( F supp  (/) ) ) )
1411, 12, 133syl 20 . . 3  |-  ( ph  ->  dom OrdIso (  _E  , 
( F supp  (/) ) )  e.  suc  dom OrdIso (  _E  ,  ( F supp  (/) ) ) )
151, 2, 3, 4, 5cantnfcl 7978 . . . . . . 7  |-  ( ph  ->  (  _E  We  ( F supp 
(/) )  /\  dom OrdIso (  _E  ,  ( F supp  (/) ) )  e.  om ) )
1615simpld 459 . . . . . 6  |-  ( ph  ->  _E  We  ( F supp  (/) ) )
174oiiso 7854 . . . . . 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 661 . . . . 5  |-  ( ph  -> OrdIso (  _E  ,  ( F supp  (/) ) )  Isom  _E  ,  _E  ( dom OrdIso (  _E  ,  ( F supp 
(/) ) ) ,  ( F supp  (/) ) ) )
19 isof1o 6117 . . . . 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 5748 . . . . 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 5725 . . . . 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 21 . . . 4  |-  ( ph  ->  (OrdIso (  _E  , 
( F supp  (/) ) )
" dom OrdIso (  _E  , 
( F supp  (/) ) ) )  =  ( F supp  (/) ) )
2322, 10eqsstrd 3490 . . 3  |-  ( ph  ->  (OrdIso (  _E  , 
( F supp  (/) ) )
" dom OrdIso (  _E  , 
( F supp  (/) ) ) )  C_  C )
241, 2, 3, 4, 5, 6, 8, 14, 9, 23cantnflt 7983 . 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 2539 1  |-  ( ph  ->  ( ( A CNF  B
) `  F )  e.  ( A  ^o  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3070    C_ wss 3428   (/)c0 3737    _E cep 4730    We wwe 4778   Oncon0 4819   suc csuc 4821   dom cdm 4940   "cima 4943   -onto->wfo 5516   -1-1-onto->wf1o 5517   ` cfv 5518    Isom wiso 5519  (class class class)co 6192    |-> cmpt2 6194   omcom 6578   supp csupp 6792  seq𝜔cseqom 7004    +o coa 7019    .o comu 7020    ^o coe 7021  OrdIsocoi 7826   CNF ccnf 7970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-supp 6793  df-recs 6934  df-rdg 6968  df-seqom 7005  df-1o 7022  df-2o 7023  df-oadd 7026  df-omul 7027  df-oexp 7028  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-fsupp 7724  df-oi 7827  df-cnf 7971
This theorem is referenced by:  cantnff  7985  cantnflem1d  7999  cnfcom3lem  8039
  Copyright terms: Public domain W3C validator