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Theorem cantnfp1 7428
Description: If  F is created by adding a single term  ( F `
 X )  =  Y to  G, where  X is larger than any element of the support of  G, then  F is also a finitely supported function and it is assigned the value  ( ( A  ^o  X )  .o  Y
)  +o  z where  z is the value of  G. (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 )
cantnfp1.4  |-  ( ph  ->  G  e.  S )
cantnfp1.5  |-  ( ph  ->  X  e.  B )
cantnfp1.6  |-  ( ph  ->  Y  e.  A )
cantnfp1.7  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  C_  X
)
cantnfp1.f  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
Assertion
Ref Expression
cantnfp1  |-  ( ph  ->  ( F  e.  S  /\  ( ( A CNF  B
) `  F )  =  ( ( ( A  ^o  X )  .o  Y )  +o  ( ( A CNF  B
) `  G )
) ) )
Distinct variable groups:    t, B    t, A    t, S    t, G    ph, t    t, Y   
t, X
Allowed substitution hint:    F( t)

Proof of Theorem cantnfp1
Dummy variables  k 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfp1.f . . . . . 6  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
2 eqeq1 2322 . . . . . . . 8  |-  ( Y  =  if ( t  =  X ,  Y ,  ( G `  t ) )  -> 
( Y  =  ( G `  t )  <-> 
if ( t  =  X ,  Y , 
( G `  t
) )  =  ( G `  t ) ) )
3 eqeq1 2322 . . . . . . . 8  |-  ( ( G `  t )  =  if ( t  =  X ,  Y ,  ( G `  t ) )  -> 
( ( G `  t )  =  ( G `  t )  <-> 
if ( t  =  X ,  Y , 
( G `  t
) )  =  ( G `  t ) ) )
4 cantnfs.3 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  On )
5 cantnfp1.5 . . . . . . . . . . . . 13  |-  ( ph  ->  X  e.  B )
6 onelon 4454 . . . . . . . . . . . . 13  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
74, 5, 6syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  On )
8 eloni 4439 . . . . . . . . . . . 12  |-  ( X  e.  On  ->  Ord  X )
9 ordirr 4447 . . . . . . . . . . . 12  |-  ( Ord 
X  ->  -.  X  e.  X )
107, 8, 93syl 18 . . . . . . . . . . 11  |-  ( ph  ->  -.  X  e.  X
)
11 fvex 5577 . . . . . . . . . . . . . 14  |-  ( G `
 X )  e. 
_V
12 dif1o 6541 . . . . . . . . . . . . . 14  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( ( G `
 X )  e. 
_V  /\  ( G `  X )  =/=  (/) ) )
1311, 12mpbiran 884 . . . . . . . . . . . . 13  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( G `  X )  =/=  (/) )
14 cantnfp1.4 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  G  e.  S )
15 cantnfs.1 . . . . . . . . . . . . . . . . . . 19  |-  S  =  dom  ( A CNF  B
)
16 cantnfs.2 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  A  e.  On )
1715, 16, 4cantnfs 7412 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
1814, 17mpbid 201 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
1918simpld 445 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G : B --> A )
20 ffn 5427 . . . . . . . . . . . . . . . 16  |-  ( G : B --> A  ->  G  Fn  B )
21 elpreima 5683 . . . . . . . . . . . . . . . 16  |-  ( G  Fn  B  ->  ( X  e.  ( `' G " ( _V  \  1o ) )  <->  ( X  e.  B  /\  ( G `  X )  e.  ( _V  \  1o ) ) ) )
2219, 20, 213syl 18 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X  e.  ( `' G " ( _V 
\  1o ) )  <-> 
( X  e.  B  /\  ( G `  X
)  e.  ( _V 
\  1o ) ) ) )
23 cantnfp1.7 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  C_  X
)
2423sseld 3213 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X  e.  ( `' G " ( _V 
\  1o ) )  ->  X  e.  X
) )
2522, 24sylbird 226 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( X  e.  B  /\  ( G `
 X )  e.  ( _V  \  1o ) )  ->  X  e.  X ) )
265, 25mpand 656 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( G `  X )  e.  ( _V  \  1o )  ->  X  e.  X
) )
2713, 26syl5bir 209 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( G `  X )  =/=  (/)  ->  X  e.  X ) )
2827necon1bd 2547 . . . . . . . . . . 11  |-  ( ph  ->  ( -.  X  e.  X  ->  ( G `  X )  =  (/) ) )
2910, 28mpd 14 . . . . . . . . . 10  |-  ( ph  ->  ( G `  X
)  =  (/) )
3029ad3antrrr 710 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  ( G `  X )  =  (/) )
31 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  t  =  X )
3231fveq2d 5567 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  ( G `  t )  =  ( G `  X ) )
33 simpllr 735 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  Y  =  (/) )
3430, 32, 333eqtr4rd 2359 . . . . . . . 8  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  Y  =  ( G `  t ) )
35 eqidd 2317 . . . . . . . 8  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  -.  t  =  X )  ->  ( G `  t )  =  ( G `  t ) )
362, 3, 34, 35ifbothda 3629 . . . . . . 7  |-  ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  ( G `
 t ) )
3736mpteq2dva 4143 . . . . . 6  |-  ( (
ph  /\  Y  =  (/) )  ->  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( G `  t ) ) )  =  ( t  e.  B  |->  ( G `  t ) ) )
381, 37syl5eq 2360 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  F  =  ( t  e.  B  |->  ( G `  t
) ) )
3919feqmptd 5613 . . . . . 6  |-  ( ph  ->  G  =  ( t  e.  B  |->  ( G `
 t ) ) )
4039adantr 451 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  G  =  ( t  e.  B  |->  ( G `  t
) ) )
4138, 40eqtr4d 2351 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  F  =  G )
4214adantr 451 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  G  e.  S )
4341, 42eqeltrd 2390 . . 3  |-  ( (
ph  /\  Y  =  (/) )  ->  F  e.  S )
44 oecl 6578 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
4516, 4, 44syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
4615, 16, 4cantnff 7420 . . . . . . . 8  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
47 ffvelrn 5701 . . . . . . . 8  |-  ( ( ( A CNF  B ) : S --> ( A  ^o  B )  /\  G  e.  S )  ->  ( ( A CNF  B
) `  G )  e.  ( A  ^o  B
) )
4846, 14, 47syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( A CNF  B
) `  G )  e.  ( A  ^o  B
) )
49 onelon 4454 . . . . . . 7  |-  ( ( ( A  ^o  B
)  e.  On  /\  ( ( A CNF  B
) `  G )  e.  ( A  ^o  B
) )  ->  (
( A CNF  B ) `
 G )  e.  On )
5045, 48, 49syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( A CNF  B
) `  G )  e.  On )
5150adantr 451 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  G
)  e.  On )
52 oa0r 6579 . . . . 5  |-  ( ( ( A CNF  B ) `
 G )  e.  On  ->  ( (/)  +o  (
( A CNF  B ) `
 G ) )  =  ( ( A CNF 
B ) `  G
) )
5351, 52syl 15 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( (/)  +o  (
( A CNF  B ) `
 G ) )  =  ( ( A CNF 
B ) `  G
) )
54 oveq2 5908 . . . . . 6  |-  ( Y  =  (/)  ->  ( ( A  ^o  X )  .o  Y )  =  ( ( A  ^o  X )  .o  (/) ) )
55 oecl 6578 . . . . . . . 8  |-  ( ( A  e.  On  /\  X  e.  On )  ->  ( A  ^o  X
)  e.  On )
5616, 7, 55syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( A  ^o  X
)  e.  On )
57 om0 6558 . . . . . . 7  |-  ( ( A  ^o  X )  e.  On  ->  (
( A  ^o  X
)  .o  (/) )  =  (/) )
5856, 57syl 15 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  X )  .o  (/) )  =  (/) )
5954, 58sylan9eqr 2370 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A  ^o  X )  .o  Y )  =  (/) )
6059oveq1d 5915 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( (
( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) )  =  (
(/)  +o  ( ( A CNF  B ) `  G
) ) )
6141fveq2d 5567 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( A CNF  B ) `  G ) )
6253, 60, 613eqtr4rd 2359 . . 3  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( ( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) ) )
6343, 62jca 518 . 2  |-  ( (
ph  /\  Y  =  (/) )  ->  ( F  e.  S  /\  (
( A CNF  B ) `
 F )  =  ( ( ( A  ^o  X )  .o  Y )  +o  (
( A CNF  B ) `
 G ) ) ) )
6416adantr 451 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  A  e.  On )
654adantr 451 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  B  e.  On )
6614adantr 451 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  G  e.  S )
675adantr 451 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  X  e.  B )
68 cantnfp1.6 . . . . 5  |-  ( ph  ->  Y  e.  A )
6968adantr 451 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  Y  e.  A )
7023adantr 451 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  ( `' G " ( _V  \  1o ) )  C_  X
)
7115, 64, 65, 66, 67, 69, 70, 1cantnfp1lem1 7425 . . 3  |-  ( (
ph  /\  Y  =/=  (/) )  ->  F  e.  S )
72 onelon 4454 . . . . . . 7  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
7316, 68, 72syl2anc 642 . . . . . 6  |-  ( ph  ->  Y  e.  On )
74 on0eln0 4484 . . . . . 6  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
7573, 74syl 15 . . . . 5  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
7675biimpar 471 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  (/)  e.  Y
)
77 eqid 2316 . . . 4  |- OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
78 eqid 2316 . . . 4  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) `  k ) )  .o  ( F `  (OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )  = seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) ) `  k ) )  .o  ( F `  (OrdIso (  _E  ,  ( `' F " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )
79 eqid 2316 . . . 4  |- OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' G " ( _V 
\  1o ) ) )
80 eqid 2316 . . . 4  |- seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) ) `  k ) )  .o  ( G `  (OrdIso (  _E  ,  ( `' G " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )  = seq𝜔 ( ( k  e. 
_V ,  z  e. 
_V  |->  ( ( ( A  ^o  (OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) ) `  k ) )  .o  ( G `  (OrdIso (  _E  ,  ( `' G " ( _V 
\  1o ) ) ) `  k ) ) )  +o  z
) ) ,  (/) )
8115, 64, 65, 66, 67, 69, 70, 1, 76, 77, 78, 79, 80cantnfp1lem3 7427 . . 3  |-  ( (
ph  /\  Y  =/=  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( ( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) ) )
8271, 81jca 518 . 2  |-  ( (
ph  /\  Y  =/=  (/) )  ->  ( F  e.  S  /\  (
( A CNF  B ) `
 F )  =  ( ( ( A  ^o  X )  .o  Y )  +o  (
( A CNF  B ) `
 G ) ) ) )
8363, 82pm2.61dane 2557 1  |-  ( ph  ->  ( F  e.  S  /\  ( ( A CNF  B
) `  F )  =  ( ( ( A  ^o  X )  .o  Y )  +o  ( ( A CNF  B
) `  G )
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701    =/= wne 2479   _Vcvv 2822    \ cdif 3183    C_ wss 3186   (/)c0 3489   ifcif 3599    e. cmpt 4114    _E cep 4340   Ord word 4428   Oncon0 4429   `'ccnv 4725   dom cdm 4726   "cima 4729    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902  seq𝜔cseqom 6501   1oc1o 6514    +o coa 6518    .o comu 6519    ^o coe 6520   Fincfn 6906  OrdIsocoi 7269   CNF ccnf 7407
This theorem is referenced by:  cantnflem1d  7435  cantnflem1  7436  cantnflem3  7438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-seqom 6502  df-1o 6521  df-2o 6522  df-oadd 6525  df-omul 6526  df-oexp 6527  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-oi 7270  df-cnf 7408
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