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Theorem cantnf0 7376
Description: The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
cantnf0.1  |-  ( ph  -> 
(/)  e.  A )
Assertion
Ref Expression
cantnf0  |-  ( ph  ->  ( ( A CNF  B
) `  ( B  X.  { (/) } ) )  =  (/) )

Proof of Theorem cantnf0
Dummy variables  k 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfs.1 . . 3  |-  S  =  dom  ( A CNF  B
)
2 cantnfs.2 . . 3  |-  ( ph  ->  A  e.  On )
3 cantnfs.3 . . 3  |-  ( ph  ->  B  e.  On )
4 eqid 2283 . . 3  |- OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) )  = OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) )
5 cantnf0.1 . . . . 5  |-  ( ph  -> 
(/)  e.  A )
6 fconst6g 5430 . . . . 5  |-  ( (/)  e.  A  ->  ( B  X.  { (/) } ) : B --> A )
75, 6syl 15 . . . 4  |-  ( ph  ->  ( B  X.  { (/)
} ) : B --> A )
8 0lt1o 6503 . . . . . . 7  |-  (/)  e.  1o
9 fconst6g 5430 . . . . . . 7  |-  ( (/)  e.  1o  ->  ( B  X.  { (/) } ) : B --> 1o )
108, 9mp1i 11 . . . . . 6  |-  ( ph  ->  ( B  X.  { (/)
} ) : B --> 1o )
11 disjdif 3526 . . . . . 6  |-  ( 1o 
i^i  ( _V  \  1o ) )  =  (/)
12 fimacnvdisj 5419 . . . . . 6  |-  ( ( ( B  X.  { (/)
} ) : B --> 1o  /\  ( 1o  i^i  ( _V  \  1o ) )  =  (/) )  -> 
( `' ( B  X.  { (/) } )
" ( _V  \  1o ) )  =  (/) )
1310, 11, 12sylancl 643 . . . . 5  |-  ( ph  ->  ( `' ( B  X.  { (/) } )
" ( _V  \  1o ) )  =  (/) )
14 0fin 7087 . . . . 5  |-  (/)  e.  Fin
1513, 14syl6eqel 2371 . . . 4  |-  ( ph  ->  ( `' ( B  X.  { (/) } )
" ( _V  \  1o ) )  e.  Fin )
161, 2, 3cantnfs 7367 . . . 4  |-  ( ph  ->  ( ( B  X.  { (/) } )  e.  S  <->  ( ( B  X.  { (/) } ) : B --> A  /\  ( `' ( B  X.  { (/) } ) "
( _V  \  1o ) )  e.  Fin ) ) )
177, 15, 16mpbir2and 888 . . 3  |-  ( ph  ->  ( B  X.  { (/)
} )  e.  S
)
18 eqid 2283 . . 3  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) )  .o  ( ( B  X.  { (/) } ) `  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )  = seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) )  .o  ( ( B  X.  { (/) } ) `  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )
191, 2, 3, 4, 17, 18cantnfval 7369 . 2  |-  ( ph  ->  ( ( A CNF  B
) `  ( B  X.  { (/) } ) )  =  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) )  .o  ( ( B  X.  { (/) } ) `  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) ) )
20 oieq2 7228 . . . . . 6  |-  ( ( `' ( B  X.  { (/) } ) "
( _V  \  1o ) )  =  (/)  -> OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) "
( _V  \  1o ) ) )  = OrdIso
(  _E  ,  (/) ) )
2113, 20syl 15 . . . . 5  |-  ( ph  -> OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) "
( _V  \  1o ) ) )  = OrdIso
(  _E  ,  (/) ) )
2221dmeqd 4881 . . . 4  |-  ( ph  ->  dom OrdIso (  _E  , 
( `' ( B  X.  { (/) } )
" ( _V  \  1o ) ) )  =  dom OrdIso (  _E  ,  (/) ) )
23 0ex 4150 . . . . . 6  |-  (/)  e.  _V
24 we0 4388 . . . . . 6  |-  _E  We  (/)
25 eqid 2283 . . . . . . 7  |- OrdIso (  _E  ,  (/) )  = OrdIso (  _E  ,  (/) )
2625oien 7253 . . . . . 6  |-  ( (
(/)  e.  _V  /\  _E  We  (/) )  ->  dom OrdIso (  _E  ,  (/) )  ~~  (/) )
2723, 24, 26mp2an 653 . . . . 5  |-  dom OrdIso (  _E  ,  (/) )  ~~  (/)
28 en0 6924 . . . . 5  |-  ( dom OrdIso (  _E  ,  (/) )  ~~  (/)  <->  dom OrdIso (  _E  ,  (/) )  =  (/) )
2927, 28mpbi 199 . . . 4  |-  dom OrdIso (  _E  ,  (/) )  =  (/)
3022, 29syl6eq 2331 . . 3  |-  ( ph  ->  dom OrdIso (  _E  , 
( `' ( B  X.  { (/) } )
" ( _V  \  1o ) ) )  =  (/) )
3130fveq2d 5529 . 2  |-  ( ph  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) )  .o  ( ( B  X.  { (/) } ) `  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  dom OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) )  =  (seq𝜔 ( ( k  e.  _V ,  z  e.  _V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) )  .o  ( ( B  X.  { (/) } ) `  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  (/) ) )
3218seqom0g 6468 . . 3  |-  ( (/)  e.  _V  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) )  .o  ( ( B  X.  { (/) } ) `  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  (/) )  =  (/) )
3323, 32mp1i 11 . 2  |-  ( ph  ->  (seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) )  .o  ( ( B  X.  { (/) } ) `  (OrdIso (  _E  ,  ( `' ( B  X.  { (/) } ) " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) ) `  (/) )  =  (/) )
3419, 31, 333eqtrd 2319 1  |-  ( ph  ->  ( ( A CNF  B
) `  ( B  X.  { (/) } ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    i^i cin 3151   (/)c0 3455   {csn 3640   class class class wbr 4023    _E cep 4303    We wwe 4351   Oncon0 4392    X. cxp 4687   `'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    ~~ cen 6860   Fincfn 6863  OrdIsocoi 7224   CNF ccnf 7362
This theorem is referenced by:  cnfcom2lem  7404
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-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-map 6774  df-en 6864  df-fin 6867  df-oi 7225  df-cnf 7363
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