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Theorem cantnffvalOLD 8082
Description: The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) Obsolete version of cantnffval 8080 as of 28-Jun-2019. Proof modified to avoid an old version of definition df-cnf 8079. (New usage 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
cantnffvalOLD  |-  ( 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
) ) )
Distinct variable groups:    f, g, h, k, z, A    B, f, g, h, k, z    S, f    ph, f
Allowed substitution hints:    ph( z, g, h, k)    S( z, g, h, k)

Proof of Theorem cantnffvalOLD
StepHypRef Expression
1 cantnffvalOLD.1 . . . 4  |-  S  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
2 elmapi 7440 . . . . . . . 8  |-  ( g  e.  ( A  ^m  B )  ->  g : B --> A )
3 ffun 5733 . . . . . . . 8  |-  ( g : B --> A  ->  Fun  g )
42, 3syl 16 . . . . . . 7  |-  ( g  e.  ( A  ^m  B )  ->  Fun  g )
5 id 22 . . . . . . 7  |-  ( g  e.  ( A  ^m  B )  ->  g  e.  ( A  ^m  B
) )
6 0ex 4577 . . . . . . . 8  |-  (/)  e.  _V
76a1i 11 . . . . . . 7  |-  ( g  e.  ( A  ^m  B )  ->  (/)  e.  _V )
8 funisfsupp 7834 . . . . . . 7  |-  ( ( Fun  g  /\  g  e.  ( A  ^m  B
)  /\  (/)  e.  _V )  ->  ( g finSupp  (/)  <->  ( g supp  (/) )  e.  Fin )
)
94, 5, 7, 8syl3anc 1228 . . . . . 6  |-  ( g  e.  ( A  ^m  B )  ->  (
g finSupp  (/)  <->  ( g supp  (/) )  e. 
Fin ) )
10 vex 3116 . . . . . . . . 9  |-  g  e. 
_V
11 suppimacnv 6912 . . . . . . . . . 10  |-  ( ( g  e.  _V  /\  (/) 
e.  _V )  ->  (
g supp  (/) )  =  ( `' g " ( _V  \  { (/) } ) ) )
12 df1o2 7142 . . . . . . . . . . . . 13  |-  1o  =  { (/) }
1312eqcomi 2480 . . . . . . . . . . . 12  |-  { (/) }  =  1o
1413difeq2i 3619 . . . . . . . . . . 11  |-  ( _V 
\  { (/) } )  =  ( _V  \  1o )
1514imaeq2i 5335 . . . . . . . . . 10  |-  ( `' g " ( _V 
\  { (/) } ) )  =  ( `' g " ( _V 
\  1o ) )
1611, 15syl6eq 2524 . . . . . . . . 9  |-  ( ( g  e.  _V  /\  (/) 
e.  _V )  ->  (
g supp  (/) )  =  ( `' g " ( _V  \  1o ) ) )
1710, 6, 16mp2an 672 . . . . . . . 8  |-  ( g supp  (/) )  =  ( `' g " ( _V  \  1o ) )
1817a1i 11 . . . . . . 7  |-  ( g  e.  ( A  ^m  B )  ->  (
g supp  (/) )  =  ( `' g " ( _V  \  1o ) ) )
1918eleq1d 2536 . . . . . 6  |-  ( g  e.  ( A  ^m  B )  ->  (
( g supp  (/) )  e. 
Fin 
<->  ( `' g "
( _V  \  1o ) )  e.  Fin ) )
209, 19bitr2d 254 . . . . 5  |-  ( g  e.  ( A  ^m  B )  ->  (
( `' g "
( _V  \  1o ) )  e.  Fin  <->  g finSupp  (/) ) )
2120rabbiia 3102 . . . 4  |-  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/) }
221, 21eqtri 2496 . . 3  |-  S  =  { g  e.  ( A  ^m  B )  |  g finSupp  (/) }
23 cantnffvalOLD.2 . . 3  |-  ( ph  ->  A  e.  On )
24 cantnffvalOLD.3 . . 3  |-  ( ph  ->  B  e.  On )
2522, 23, 24cantnffval 8080 . 2  |-  ( 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 ) ) )
26 vex 3116 . . . . . . . 8  |-  f  e. 
_V
2726, 6pm3.2i 455 . . . . . . 7  |-  ( f  e.  _V  /\  (/)  e.  _V )
28 suppimacnv 6912 . . . . . . 7  |-  ( ( f  e.  _V  /\  (/) 
e.  _V )  ->  (
f supp  (/) )  =  ( `' f " ( _V  \  { (/) } ) ) )
2927, 28mp1i 12 . . . . . 6  |-  ( ph  ->  ( f supp  (/) )  =  ( `' f "
( _V  \  { (/)
} ) ) )
3013a1i 11 . . . . . . . 8  |-  ( ph  ->  { (/) }  =  1o )
3130difeq2d 3622 . . . . . . 7  |-  ( ph  ->  ( _V  \  { (/)
} )  =  ( _V  \  1o ) )
3231imaeq2d 5337 . . . . . 6  |-  ( ph  ->  ( `' f "
( _V  \  { (/)
} ) )  =  ( `' f "
( _V  \  1o ) ) )
3329, 32eqtrd 2508 . . . . 5  |-  ( ph  ->  ( f supp  (/) )  =  ( `' f "
( _V  \  1o ) ) )
34 oieq2 7938 . . . . 5  |-  ( ( f supp  (/) )  =  ( `' f " ( _V  \  1o ) )  -> OrdIso (  _E  ,  ( f supp  (/) ) )  = OrdIso
(  _E  ,  ( `' f " ( _V  \  1o ) ) ) )
3533, 34syl 16 . . . 4  |-  ( ph  -> OrdIso (  _E  ,  ( f supp  (/) ) )  = OrdIso
(  _E  ,  ( `' f " ( _V  \  1o ) ) ) )
3635csbeq1d 3442 . . 3  |-  ( ph  ->  [_OrdIso (  _E  , 
( f supp  (/) ) )  /  h ]_ (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  ( h `  k ) )  .o  ( f `  (
h `  k )
) )  +o  z
) ) ,  (/) ) `  dom  h )  =  [_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
) )
3736mpteq2dv 4534 . 2  |-  ( ph  ->  ( 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
) )  =  ( 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
) ) )
3825, 37eqtrd 2508 1  |-  ( 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
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113   [_csb 3435    \ cdif 3473   (/)c0 3785   {csn 4027   class class class wbr 4447    |-> cmpt 4505    _E cep 4789   Oncon0 4878   `'ccnv 4998   dom cdm 4999   "cima 5002   Fun wfun 5582   -->wf 5584   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   supp csupp 6901  seq𝜔cseqom 7112   1oc1o 7123    +o coa 7127    .o comu 7128    ^o coe 7129    ^m cmap 7420   Fincfn 7516   finSupp cfsupp 7829  OrdIsocoi 7934   CNF ccnf 8078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-seqom 7113  df-1o 7130  df-map 7422  df-fsupp 7830  df-oi 7935  df-cnf 8079
This theorem is referenced by:  cantnfdmOLD  8083  cantnfvalOLD  8117
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