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Theorem cantnff 7391
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 2804 . . . . . 6  |-  f  e. 
_V
21cnvex 5225 . . . . 5  |-  `' f  e.  _V
3 imaexg 5042 . . . . 5  |-  ( `' f  e.  _V  ->  ( `' f " ( _V  \  1o ) )  e.  _V )
4 eqid 2296 . . . . . 6  |- OrdIso (  _E  ,  ( `' f
" ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' f " ( _V  \  1o ) ) )
54oiexg 7266 . . . . 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 5555 . . . 4  |-  (seq𝜔 ( ( k  e.  _V , 
z  e.  _V  |->  ( ( ( A  ^o  ( h `  k
) )  .o  (
f `  ( h `  k ) ) )  +o  z ) ) ,  (/) ) `  dom  h )  e.  _V
86, 7csbex 3105 . . 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 2296 . . . 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 7380 . . 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 7381 . . . . 5  |-  ( ph  ->  dom  ( A CNF  B
)  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
1614, 15syl5eq 2340 . . . 4  |-  ( ph  ->  S  =  { g  e.  ( A  ^m  B )  |  ( `' g " ( _V  \  1o ) )  e.  Fin } )
17 mpteq1 4116 . . . 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 2331 . 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 2296 . . . . . . . 8  |- OrdIso (  _E  ,  ( `' x " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' x " ( _V 
\  1o ) ) )
23 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  S )
24 eqid 2296 . . . . . . . 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 7385 . . . . . . 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 2804 . . . . . . . . . . . . 13  |-  x  e. 
_V
2827cnvex 5225 . . . . . . . . . . . 12  |-  `' x  e.  _V
29 imaexg 5042 . . . . . . . . . . . 12  |-  ( `' x  e.  _V  ->  ( `' x " ( _V 
\  1o ) )  e.  _V )
3028, 29ax-mp 8 . . . . . . . . . . 11  |-  ( `' x " ( _V 
\  1o ) )  e.  _V
3114, 20, 21, 22, 23cantnfcl 7384 . . . . . . . . . . . 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 7269 . . . . . . . . . . 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 5049 . . . . . . . . . . . 12  |-  ( `' x " ( _V 
\  1o ) ) 
C_  dom  x
3714, 11, 12cantnfs 7383 . . . . . . . . . . . . . 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 5409 . . . . . . . . . . . . 13  |-  ( x : B --> A  ->  dom  x  =  B )
4038, 39syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  S )  ->  dom  x  =  B )
4136, 40syl5sseq 3239 . . . . . . . . . . 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 5393 . . . . . . . . . . . . . 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 5442 . . . . . . . . . . . 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 3499 . . . . . . . . . 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 4063 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  dom OrdIso (  _E  ,  ( `' x " ( _V 
\  1o ) ) )  ~~  (/) )
52 en0 6940 . . . . . . . 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 5545 . . . . . 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 4166 . . . . . . 7  |-  (/)  e.  _V
5624seqom0g 6484 . . . . . . 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 2332 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  (
( A CNF  B ) `
 x )  =  (/) )
59 el1o 6514 . . . . 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 5890 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  ( A  ^o  (/) ) )
6220adantr 451 . . . . . 6  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  A  e.  On )
63 oe0 6537 . . . . . 6  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
6462, 63syl 15 . . . . 5  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  (/) )  =  1o )
6561, 64eqtrd 2328 . . . 4  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =  (/) )  ->  ( A  ^o  B )  =  1o )
6660, 65eleqtrrd 2373 . . 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 4463 . . . . . 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 7390 . . 3  |-  ( ( ( ph  /\  x  e.  S )  /\  A  =/=  (/) )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
7566, 74pm2.61dane 2537 . 2  |-  ( (
ph  /\  x  e.  S )  ->  (
( A CNF  B ) `
 x )  e.  ( A  ^o  B
) )
769, 19, 75fmpt2d 5704 1  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801   [_csb 3094    \ cdif 3162    C_ wss 3165   (/)c0 3468   class class class wbr 4039    e. cmpt 4093    _E cep 4319    We wwe 4367   Oncon0 4408   omcom 4672   `'ccnv 4704   dom cdm 4705   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876  seq𝜔cseqom 6475   1oc1o 6488    +o coa 6492    .o comu 6493    ^o coe 6494    ^m cmap 6788    ~~ cen 6876   Fincfn 6879  OrdIsocoi 7240   CNF ccnf 7378
This theorem is referenced by:  cantnfp1  7399  cantnflem1  7407  cantnflem3  7409  cantnflem4  7410  cantnf  7411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-seqom 6476  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-oexp 6501  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-cnf 7379
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