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Theorem cantnff 7629
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 2959 . . . . . 6  |-  f  e. 
_V
21cnvex 5406 . . . . 5  |-  `' f  e.  _V
3 imaexg 5217 . . . . 5  |-  ( `' f  e.  _V  ->  ( `' f " ( _V  \  1o ) )  e.  _V )
4 eqid 2436 . . . . . 6  |- OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' f " ( _V  \  1o ) ) )
54oiexg 7504 . . . . 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 5742 . . . 4  |-  (seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h )  e.  _V
86, 7csbex 3262 . . 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 2436 . . . 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 7618 . . 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 7619 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
1614, 15syl5eq 2480 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
1716mpteq1d 4290 . . 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 2471 . 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 2436 . . . . . . . 8  |- OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' x " ( _V 
\  1o ) ) )
22 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
23 eqid 2436 . . . . . . . 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 7623 . . . . . . 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 2959 . . . . . . . . . . . . 13  |-  x  e. 
_V
2726cnvex 5406 . . . . . . . . . . . 12  |-  `' x  e.  _V
28 imaexg 5217 . . . . . . . . . . . 12  |-  ( `' x  e.  _V  ->  ( `' x " ( _V 
\  1o ) )  e.  _V )
2927, 28ax-mp 8 . . . . . . . . . . 11  |-  ( `' x " ( _V 
\  1o ) )  e.  _V
3014, 19, 20, 21, 22cantnfcl 7622 . . . . . . . . . . . 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 7507 . . . . . . . . . . 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 5224 . . . . . . . . . . . 12  |-  ( `' x " ( _V 
\  1o ) ) 
C_  dom  x
3614, 11, 12cantnfs 7621 . . . . . . . . . . . . . 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 5595 . . . . . . . . . . . . 13  |-  ( x : B --> A  ->  dom  x  =  B )
3937, 38syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  S )  ->  dom  x  =  B )
4035, 39syl5sseq 3396 . . . . . . . . . . 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 5578 . . . . . . . . . . . . . 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 5628 . . . . . . . . . . . 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 3659 . . . . . . . . . 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 4236 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) )  ~~  (/) )
51 en0 7170 . . . . . . . 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 5732 . . . . . 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 4339 . . . . . . 7  |-  (/)  e.  _V
5523seqom0g 6713 . . . . . . 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 2472 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  =  (/) )
58 el1o 6743 . . . . 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 6097 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  ( A  ^o  (/) ) )
6119adantr 452 . . . . . 6  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  A  e.  On )
62 oe0 6766 . . . . . 6  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
6361, 62syl 16 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  (/) )  =  1o )
6460, 63eqtrd 2468 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  1o )
6559, 64eleqtrrd 2513 . . 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 4636 . . . . . 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 7628 . . 3  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
7465, 73pm2.61dane 2682 . 2  |-  ( (
ph  /\  x  e.  S )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
759, 18, 74fmpt2d 5898 1  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   {crab 2709   _Vcvv 2956   [_csb 3251    \ cdif 3317    C_ wss 3320   (/)c0 3628   class class class wbr 4212    e. cmpt 4266    _E cep 4492    We wwe 4540   Oncon0 4581   omcom 4845   `'ccnv 4877   dom cdm 4878   "cima 4881   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083  seq𝜔cseqom 6704   1oc1o 6717    +o coa 6721    .o comu 6722    ^o coe 6723    ^m cmap 7018    ~~ cen 7106   Fincfn 7109  OrdIsocoi 7478   CNF ccnf 7616
This theorem is referenced by:  cantnfp1  7637  cantnflem1  7645  cantnflem3  7647  cantnflem4  7648  cantnf  7649
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-seqom 6705  df-1o 6724  df-2o 6725  df-oadd 6728  df-omul 6729  df-oexp 6730  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-cnf 7617
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