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Theorem cantnff 7375
Description: The CNF function is a function from finitely supported functions from  B to  A, to the ordinal exponential  A  ^o  B. (Contributed by Mario Carneiro, 28-May-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
Assertion
Ref Expression
cantnff  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )

Proof of Theorem cantnff
Dummy variables  f 
g  h  k  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . . 6  |-  f  e. 
_V
21cnvex 5209 . . . . 5  |-  `' f  e.  _V
3 imaexg 5026 . . . . 5  |-  ( `' f  e.  _V  ->  ( `' f " ( _V  \  1o ) )  e.  _V )
4 eqid 2283 . . . . . 6  |- OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' f " ( _V  \  1o ) ) )
54oiexg 7250 . . . . 5  |-  ( ( `' f " ( _V  \  1o ) )  e.  _V  -> OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  e. 
_V )
62, 3, 5mp2b 9 . . . 4  |- OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  e. 
_V
7 fvex 5539 . . . 4  |-  (seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h )  e.  _V
86, 7csbex 3092 . . 3  |-  [_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
98a1i 10 . 2  |-  ( (
ph  /\  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 )
10 eqid 2283 . . . 4  |-  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  =  { g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin }
11 cantnfs.2 . . . 4  |-  ( ph  ->  A  e.  On )
12 cantnfs.3 . . . 4  |-  ( ph  ->  B  e.  On )
1310, 11, 12cantnffval 7364 . . 3  |-  ( ph  ->  ( A CNF  B )  =  ( f  e. 
{ g  e.  ( A  ^m  B )  |  ( `' g
" ( _V  \  1o ) )  e.  Fin } 
|->  [_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
) ) )
14 cantnfs.1 . . . . 5  |-  S  =  dom  ( A CNF  B
)
1510, 11, 12cantnfdm 7365 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
1614, 15syl5eq 2327 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
17 mpteq1 4100 . . . 4  |-  ( S  =  { g  e.  ( A  ^m  B
)  |  ( `' g " ( _V 
\  1o ) )  e.  Fin }  ->  ( 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
) )  =  ( f  e.  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  |->  [_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
) ) )
1816, 17syl 15 . . 3  |-  ( ph  ->  ( 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
) )  =  ( f  e.  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin }  |->  [_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
) ) )
1913, 18eqtr4d 2318 . 2  |-  ( 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
) ) )
2011adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  On )
2112adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  B  e.  On )
22 eqid 2283 . . . . . . . 8  |- OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' x " ( _V 
\  1o ) ) )
23 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
24 eqid 2283 . . . . . . . 8  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) ) `  k ) )  .o  ( x `  (OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )  = seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) ) `  k ) )  .o  ( x `  (OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )
2514, 20, 21, 22, 23, 24cantnfval 7369 . . . . . . 7  |-  ( (
ph  /\  x  e.  S )  ->  (
( A CNF  B ) `
 x )  =  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) ) `  k ) )  .o  ( x `  (OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) ) ) )
2625adantr 451 . . . . . 6  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  =  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) ) `  k ) )  .o  ( x `  (OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) ) ) )
27 vex 2791 . . . . . . . . . . . . 13  |-  x  e. 
_V
2827cnvex 5209 . . . . . . . . . . . 12  |-  `' x  e.  _V
29 imaexg 5026 . . . . . . . . . . . 12  |-  ( `' x  e.  _V  ->  ( `' x " ( _V 
\  1o ) )  e.  _V )
3028, 29ax-mp 8 . . . . . . . . . . 11  |-  ( `' x " ( _V 
\  1o ) )  e.  _V
3114, 20, 21, 22, 23cantnfcl 7368 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  S )  ->  (  _E  We  ( `' x " ( _V  \  1o ) )  /\  dom OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) )  e.  om )
)
3231simpld 445 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  _E  We  ( `' x "
( _V  \  1o ) ) )
3322oien 7253 . . . . . . . . . . 11  |-  ( ( ( `' x "
( _V  \  1o ) )  e.  _V  /\  _E  We  ( `' x " ( _V 
\  1o ) ) )  ->  dom OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) )  ~~  ( `' x " ( _V 
\  1o ) ) )
3430, 32, 33sylancr 644 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  S )  ->  dom OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) )  ~~  ( `' x " ( _V 
\  1o ) ) )
3534adantr 451 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) )  ~~  ( `' x " ( _V 
\  1o ) ) )
36 cnvimass 5033 . . . . . . . . . . . 12  |-  ( `' x " ( _V 
\  1o ) ) 
C_  dom  x
3714, 11, 12cantnfs 7367 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  S  <->  ( x : B --> A  /\  ( `' x " ( _V 
\  1o ) )  e.  Fin ) ) )
3837simprbda 606 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  S )  ->  x : B --> A )
39 fdm 5393 . . . . . . . . . . . . 13  |-  ( x : B --> A  ->  dom  x  =  B )
4038, 39syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  S )  ->  dom  x  =  B )
4136, 40syl5sseq 3226 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  ( `' x " ( _V 
\  1o ) ) 
C_  B )
4241adantr 451 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( `' x " ( _V 
\  1o ) ) 
C_  B )
43 feq3 5377 . . . . . . . . . . . . . 14  |-  ( A  =  (/)  ->  ( x : B --> A  <->  x : B
--> (/) ) )
4438, 43syl5ibcom 211 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  S )  ->  ( A  =  (/)  ->  x : B --> (/) ) )
4544imp 418 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  x : B --> (/) )
46 f00 5426 . . . . . . . . . . . 12  |-  ( x : B --> (/)  <->  ( x  =  (/)  /\  B  =  (/) ) )
4745, 46sylib 188 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
x  =  (/)  /\  B  =  (/) ) )
4847simprd 449 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  B  =  (/) )
49 sseq0 3486 . . . . . . . . . 10  |-  ( ( ( `' x "
( _V  \  1o ) )  C_  B  /\  B  =  (/) )  -> 
( `' x "
( _V  \  1o ) )  =  (/) )
5042, 48, 49syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( `' x " ( _V 
\  1o ) )  =  (/) )
5135, 50breqtrd 4047 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) )  ~~  (/) )
52 en0 6924 . . . . . . . 8  |-  ( dom OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) )  ~~  (/)  <->  dom OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) )  =  (/) )
5351, 52sylib 188 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) )  =  (/) )
5453fveq2d 5529 . . . . . 6  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) ) `  k ) )  .o  ( x `  (OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) ) )  =  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) ) `  k ) )  .o  ( x `  (OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  (/) ) )
55 0ex 4150 . . . . . . 7  |-  (/)  e.  _V
5624seqom0g 6468 . . . . . . 7  |-  ( (/)  e.  _V  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) ) `  k ) )  .o  ( x `  (OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  (/) )  =  (/) )
5755, 56mp1i 11 . . . . . 6  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (seq𝜔 (
( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) ) `  k ) )  .o  ( x `  (OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  (/) )  =  (/) )
5826, 54, 573eqtrd 2319 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  =  (/) )
59 el1o 6498 . . . . 5  |-  ( ( ( A CNF  B ) `
 x )  e.  1o  <->  ( ( A CNF 
B ) `  x
)  =  (/) )
6058, 59sylibr 203 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  e.  1o )
6148oveq2d 5874 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  ( A  ^o  (/) ) )
6220adantr 451 . . . . . 6  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  A  e.  On )
63 oe0 6521 . . . . . 6  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
6462, 63syl 15 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  (/) )  =  1o )
6561, 64eqtrd 2315 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  1o )
6660, 65eleqtrrd 2360 . . 3  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
6720adantr 451 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  A  e.  On )
6821adantr 451 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  B  e.  On )
6923adantr 451 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  x  e.  S )
70 on0eln0 4447 . . . . . 6  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
7120, 70syl 15 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
7271biimpar 471 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (/)  e.  A
)
7341adantr 451 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  ( `' x " ( _V 
\  1o ) ) 
C_  B )
7414, 67, 68, 69, 72, 68, 73cantnflt2 7374 . . 3  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
7566, 74pm2.61dane 2524 . 2  |-  ( (
ph  /\  x  e.  S )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
769, 19, 75fmpt2d 5688 1  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788   [_csb 3081    \ cdif 3149    C_ wss 3152   (/)c0 3455   class class class wbr 4023    e. cmpt 4077    _E cep 4303    We wwe 4351   Oncon0 4392   omcom 4656   `'ccnv 4688   dom cdm 4689   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860  seq𝜔cseqom 6459   1oc1o 6472    +o coa 6476    .o comu 6477    ^o coe 6478    ^m cmap 6772    ~~ cen 6860   Fincfn 6863  OrdIsocoi 7224   CNF ccnf 7362
This theorem is referenced by:  cantnfp1  7383  cantnflem1  7391  cantnflem3  7393  cantnflem4  7394  cantnf  7395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-oexp 6485  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-cnf 7363
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