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

Theorem cantnfdm 7971
Description: The domain of the Cantor normal form function (in later lemmas we will use  dom  ( A CNF 
B ) to abbreviate "the set of finitely supported functions from  B to  A"). (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
Hypotheses
Ref Expression
cantnffval.s  |-  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/) }
cantnffval.a  |-  ( ph  ->  A  e.  On )
cantnffval.b  |-  ( ph  ->  B  e.  On )
Assertion
Ref Expression
cantnfdm  |-  ( ph  ->  dom  ( A CNF  B
)  =  S )
Distinct variable groups:    A, g    B, g
Allowed substitution hints:    ph( g)    S( g)

Proof of Theorem cantnfdm
Dummy variables  f  h  k  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnffval.s . . . 4  |-  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/) }
2 cantnffval.a . . . 4  |-  ( ph  ->  A  e.  On )
3 cantnffval.b . . . 4  |-  ( ph  ->  B  e.  On )
41, 2, 3cantnffval 7970 . . 3  |-  ( ph  ->  ( A CNF  B )  =  ( f  e.  S  |->  [_OrdIso (  _E  , 
( f supp  (/) ) )  /  h ]_ (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) ) `  dom  h ) ) )
54dmeqd 5140 . 2  |-  ( ph  ->  dom  ( A CNF  B
)  =  dom  (
f  e.  S  |->  [_OrdIso (  _E  ,  ( f supp  (/) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) ) )
6 fvex 5799 . . . . 5  |-  (seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h )  e.  _V
76csbex 4523 . . . 4  |-  [_OrdIso (  _E  ,  ( f supp  (/) ) )  /  h ]_ (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) ) `  dom  h )  e.  _V
87rgenw 2891 . . 3  |-  A. f  e.  S  [_OrdIso (  _E  ,  ( f supp  (/) ) )  /  h ]_ (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) ) `  dom  h )  e.  _V
9 dmmptg 5433 . . 3  |-  ( A. f  e.  S  [_OrdIso (  _E  ,  ( f supp  (/) ) )  /  h ]_ (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) ) `  dom  h )  e.  _V  ->  dom  ( f  e.  S  |-> 
[_OrdIso (  _E  ,  ( f supp  (/) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) )  =  S )
108, 9ax-mp 5 . 2  |-  dom  (
f  e.  S  |->  [_OrdIso (  _E  ,  ( f supp  (/) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) )  =  S
115, 10syl6eq 2508 1  |-  ( ph  ->  dom  ( A CNF  B
)  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   A.wral 2795   {crab 2799   _Vcvv 3068   [_csb 3386   (/)c0 3735   class class class wbr 4390    |-> cmpt 4448    _E cep 4728   Oncon0 4817   dom cdm 4938   ` cfv 5516  (class class class)co 6190    |-> cmpt2 6192   supp csupp 6790  seq𝜔cseqom 7002    +o coa 7017    .o comu 7018    ^o coe 7019    ^m cmap 7314   finSupp cfsupp 7721  OrdIsocoi 7824   CNF ccnf 7968
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-recs 6932  df-rdg 6966  df-seqom 7003  df-cnf 7969
This theorem is referenced by:  cantnfs  7975  cantnfval  7977  cantnff  7983  oemapso  7991  wemapwe  8029  oef1o  8031
  Copyright terms: Public domain W3C validator