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Theorem cantnfdm 7611
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.)
Hypotheses
Ref Expression
cantnffval.1  |-  S  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
cantnffval.2  |-  ( ph  ->  A  e.  On )
cantnffval.3  |-  ( 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.1 . . . 4  |-  S  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
2 cantnffval.2 . . . 4  |-  ( ph  ->  A  e.  On )
3 cantnffval.3 . . . 4  |-  ( ph  ->  B  e.  On )
41, 2, 3cantnffval 7610 . . 3  |-  ( ph  ->  ( A CNF  B )  =  ( f  e.  S  |->  [_OrdIso (  _E  , 
( `' f "
( _V  \  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) ) )
54dmeqd 5064 . 2  |-  ( ph  ->  dom  ( A CNF  B
)  =  dom  (
f  e.  S  |->  [_OrdIso (  _E  ,  ( `' f " ( _V 
\  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) ) )
6 vex 2951 . . . . . . 7  |-  f  e. 
_V
76cnvex 5398 . . . . . 6  |-  `' f  e.  _V
8 imaexg 5209 . . . . . 6  |-  ( `' f  e.  _V  ->  ( `' f " ( _V  \  1o ) )  e.  _V )
9 eqid 2435 . . . . . . 7  |- OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' f " ( _V  \  1o ) ) )
109oiexg 7496 . . . . . 6  |-  ( ( `' f " ( _V  \  1o ) )  e.  _V  -> OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  e. 
_V )
117, 8, 10mp2b 10 . . . . 5  |- OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  e. 
_V
12 fvex 5734 . . . . 5  |-  (seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h )  e.  _V
1311, 12csbex 3254 . . . 4  |-  [_OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
)  e.  _V
1413rgenw 2765 . . 3  |-  A. f  e.  S  [_OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
)  e.  _V
15 dmmptg 5359 . . 3  |-  ( A. f  e.  S  [_OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  /  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 " ( _V  \  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) )  =  S )
1614, 15ax-mp 8 . 2  |-  dom  (
f  e.  S  |->  [_OrdIso (  _E  ,  ( `' f " ( _V 
\  1o ) ) )  /  h ]_ (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  ( h `
 k ) )  .o  ( f `  ( h `  k
) ) )  +o  z ) ) ,  (/) ) `  dom  h
) )  =  S
175, 16syl6eq 2483 1  |-  ( ph  ->  dom  ( A CNF  B
)  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948   [_csb 3243    \ cdif 3309   (/)c0 3620    e. cmpt 4258    _E cep 4484   Oncon0 4573   `'ccnv 4869   dom cdm 4870   "cima 4873   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075  seq𝜔cseqom 6696   1oc1o 6709    +o coa 6713    .o comu 6714    ^o coe 6715    ^m cmap 7010   Fincfn 7101  OrdIsocoi 7470   CNF ccnf 7608
This theorem is referenced by:  cantnfs  7613  cantnfval  7615  cantnff  7621  oemapso  7630  wemapwe  7646  oef1o  7647
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-seqom 6697  df-oi 7471  df-cnf 7609
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