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Theorem cantnfdm 8072
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 8071 . . 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 5194 . 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 5858 . . . . 5  |-  (seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h )  e.  _V
76csbex 4572 . . . 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 2815 . . 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 5487 . . 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 2511 1  |-  ( ph  ->  dom  ( A CNF  B
)  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   _Vcvv 3106   [_csb 3420   (/)c0 3783   class class class wbr 4439    |-> cmpt 4497    _E cep 4778   Oncon0 4867   dom cdm 4988   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   supp csupp 6891  seq𝜔cseqom 7104    +o coa 7119    .o comu 7120    ^o coe 7121    ^m cmap 7412   finSupp cfsupp 7821  OrdIsocoi 7926   CNF ccnf 8069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-recs 7034  df-rdg 7068  df-seqom 7105  df-cnf 8070
This theorem is referenced by:  cantnfs  8076  cantnfval  8078  cantnff  8084  oemapso  8092  wemapwe  8130  oef1o  8132
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