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Theorem cantnff 7585
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 2919 . . . . . 6  |-  f  e. 
_V
21cnvex 5365 . . . . 5  |-  `' f  e.  _V
3 imaexg 5176 . . . . 5  |-  ( `' f  e.  _V  ->  ( `' f " ( _V  \  1o ) )  e.  _V )
4 eqid 2404 . . . . . 6  |- OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' f " ( _V  \  1o ) ) )
54oiexg 7460 . . . . 5  |-  ( ( `' f " ( _V  \  1o ) )  e.  _V  -> OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  e. 
_V )
62, 3, 5mp2b 10 . . . 4  |- OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  e. 
_V
7 fvex 5701 . . . 4  |-  (seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h )  e.  _V
86, 7csbex 3222 . . 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 11 . 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 2404 . . . 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 7574 . . 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 7575 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
1614, 15syl5eq 2448 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
1716mpteq1d 4250 . . 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
) ) )
1813, 17eqtr4d 2439 . 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
) ) )
1911adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  On )
2012adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  B  e.  On )
21 eqid 2404 . . . . . . . 8  |- OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' x " ( _V 
\  1o ) ) )
22 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
23 eqid 2404 . . . . . . . 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
) ) ,  (/) )
2414, 19, 20, 21, 22, 23cantnfval 7579 . . . . . . 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 ) ) ) ) )
2524adantr 452 . . . . . 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 ) ) ) ) )
26 vex 2919 . . . . . . . . . . . . 13  |-  x  e. 
_V
2726cnvex 5365 . . . . . . . . . . . 12  |-  `' x  e.  _V
28 imaexg 5176 . . . . . . . . . . . 12  |-  ( `' x  e.  _V  ->  ( `' x " ( _V 
\  1o ) )  e.  _V )
2927, 28ax-mp 8 . . . . . . . . . . 11  |-  ( `' x " ( _V 
\  1o ) )  e.  _V
3014, 19, 20, 21, 22cantnfcl 7578 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  S )  ->  (  _E  We  ( `' x " ( _V  \  1o ) )  /\  dom OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) )  e.  om )
)
3130simpld 446 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  _E  We  ( `' x "
( _V  \  1o ) ) )
3221oien 7463 . . . . . . . . . . 11  |-  ( ( ( `' x "
( _V  \  1o ) )  e.  _V  /\  _E  We  ( `' x " ( _V 
\  1o ) ) )  ->  dom OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) )  ~~  ( `' x " ( _V 
\  1o ) ) )
3329, 31, 32sylancr 645 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  S )  ->  dom OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) )  ~~  ( `' x " ( _V 
\  1o ) ) )
3433adantr 452 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) )  ~~  ( `' x " ( _V 
\  1o ) ) )
35 cnvimass 5183 . . . . . . . . . . . 12  |-  ( `' x " ( _V 
\  1o ) ) 
C_  dom  x
3614, 11, 12cantnfs 7577 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( x  e.  S  <->  ( x : B --> A  /\  ( `' x " ( _V 
\  1o ) )  e.  Fin ) ) )
3736simprbda 607 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  S )  ->  x : B --> A )
38 fdm 5554 . . . . . . . . . . . . 13  |-  ( x : B --> A  ->  dom  x  =  B )
3937, 38syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  S )  ->  dom  x  =  B )
4035, 39syl5sseq 3356 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  ( `' x " ( _V 
\  1o ) ) 
C_  B )
4140adantr 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( `' x " ( _V 
\  1o ) ) 
C_  B )
42 feq3 5537 . . . . . . . . . . . . . 14  |-  ( A  =  (/)  ->  ( x : B --> A  <->  x : B
--> (/) ) )
4337, 42syl5ibcom 212 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  S )  ->  ( A  =  (/)  ->  x : B --> (/) ) )
4443imp 419 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  x : B --> (/) )
45 f00 5587 . . . . . . . . . . . 12  |-  ( x : B --> (/)  <->  ( x  =  (/)  /\  B  =  (/) ) )
4644, 45sylib 189 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
x  =  (/)  /\  B  =  (/) ) )
4746simprd 450 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  B  =  (/) )
48 sseq0 3619 . . . . . . . . . 10  |-  ( ( ( `' x "
( _V  \  1o ) )  C_  B  /\  B  =  (/) )  -> 
( `' x "
( _V  \  1o ) )  =  (/) )
4941, 47, 48syl2anc 643 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( `' x " ( _V 
\  1o ) )  =  (/) )
5034, 49breqtrd 4196 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) )  ~~  (/) )
51 en0 7129 . . . . . . . 8  |-  ( dom OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) )  ~~  (/)  <->  dom OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) )  =  (/) )
5250, 51sylib 189 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) )  =  (/) )
5352fveq2d 5691 . . . . . 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
) ) ,  (/) ) `  (/) ) )
54 0ex 4299 . . . . . . 7  |-  (/)  e.  _V
5523seqom0g 6672 . . . . . . 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
) ) ,  (/) ) `  (/) )  =  (/) )
5654, 55mp1i 12 . . . . . 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
) ) ,  (/) ) `  (/) )  =  (/) )
5725, 53, 563eqtrd 2440 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  =  (/) )
58 el1o 6702 . . . . 5  |-  ( ( ( A CNF  B ) `
 x )  e.  1o  <->  ( ( A CNF 
B ) `  x
)  =  (/) )
5957, 58sylibr 204 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  e.  1o )
6047oveq2d 6056 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  ( A  ^o  (/) ) )
6119adantr 452 . . . . . 6  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  A  e.  On )
62 oe0 6725 . . . . . 6  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
6361, 62syl 16 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  (/) )  =  1o )
6460, 63eqtrd 2436 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  1o )
6559, 64eleqtrrd 2481 . . 3  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
6619adantr 452 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  A  e.  On )
6720adantr 452 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  B  e.  On )
6822adantr 452 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  x  e.  S )
69 on0eln0 4596 . . . . . 6  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
7019, 69syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  ( (/) 
e.  A  <->  A  =/=  (/) ) )
7170biimpar 472 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (/)  e.  A
)
7240adantr 452 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  ( `' x " ( _V 
\  1o ) ) 
C_  B )
7314, 66, 67, 68, 71, 67, 72cantnflt2 7584 . . 3  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
7465, 73pm2.61dane 2645 . 2  |-  ( (
ph  /\  x  e.  S )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
759, 18, 74fmpt2d 5857 1  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   {crab 2670   _Vcvv 2916   [_csb 3211    \ cdif 3277    C_ wss 3280   (/)c0 3588   class class class wbr 4172    e. cmpt 4226    _E cep 4452    We wwe 4500   Oncon0 4541   omcom 4804   `'ccnv 4836   dom cdm 4837   "cima 4840   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042  seq𝜔cseqom 6663   1oc1o 6676    +o coa 6680    .o comu 6681    ^o coe 6682    ^m cmap 6977    ~~ cen 7065   Fincfn 7068  OrdIsocoi 7434   CNF ccnf 7572
This theorem is referenced by:  cantnfp1  7593  cantnflem1  7601  cantnflem3  7603  cantnflem4  7604  cantnf  7605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-seqom 6664  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-oexp 6689  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-cnf 7573
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