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Theorem cantnfdmOLD 7996
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.) Obsolete version of cantnfdm 7994 as of 28-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
cantnffvalOLD.1  |-  S  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
cantnffvalOLD.2  |-  ( ph  ->  A  e.  On )
cantnffvalOLD.3  |-  ( ph  ->  B  e.  On )
Assertion
Ref Expression
cantnfdmOLD  |-  ( ph  ->  dom  ( A CNF  B
)  =  S )
Distinct variable groups:    A, g    B, g
Allowed substitution hints:    ph( g)    S( g)

Proof of Theorem cantnfdmOLD
Dummy variables  f  h  k  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnffvalOLD.1 . . . 4  |-  S  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
2 cantnffvalOLD.2 . . . 4  |-  ( ph  ->  A  e.  On )
3 cantnffvalOLD.3 . . . 4  |-  ( ph  ->  B  e.  On )
41, 2, 3cantnffvalOLD 7995 . . 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 5118 . 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 fvex 5784 . . . . 5  |-  (seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h )  e.  _V
76csbex 4500 . . . 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
87rgenw 2743 . . 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
9 dmmptg 5412 . . 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 )
108, 9ax-mp 5 . 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
115, 10syl6eq 2439 1  |-  ( ph  ->  dom  ( A CNF  B
)  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826   A.wral 2732   {crab 2736   _Vcvv 3034   [_csb 3348    \ cdif 3386   (/)c0 3711    |-> cmpt 4425    _E cep 4703   Oncon0 4792   `'ccnv 4912   dom cdm 4913   "cima 4916   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198  seq𝜔cseqom 7030   1oc1o 7041    +o coa 7045    .o comu 7046    ^o coe 7047    ^m cmap 7338   Fincfn 7435  OrdIsocoi 7849   CNF ccnf 7991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-seqom 7031  df-1o 7048  df-map 7340  df-fsupp 7745  df-oi 7850  df-cnf 7992
This theorem is referenced by:  cantnfsOLD  8028  cantnfvalOLD  8030  wemapweOLD  8053  oef1oOLD  8055
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