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Theorem cantnfp1 7639
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 2444 . . . . . . . 8  |-  ( Y  =  if ( t  =  X ,  Y ,  ( G `  t ) )  -> 
( Y  =  ( G `  t )  <-> 
if ( t  =  X ,  Y , 
( G `  t
) )  =  ( G `  t ) ) )
3 eqeq1 2444 . . . . . . . 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 4608 . . . . . . . . . . . . 13  |-  ( ( B  e.  On  /\  X  e.  B )  ->  X  e.  On )
74, 5, 6syl2anc 644 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  On )
8 eloni 4593 . . . . . . . . . . . 12  |-  ( X  e.  On  ->  Ord  X )
9 ordirr 4601 . . . . . . . . . . . 12  |-  ( Ord 
X  ->  -.  X  e.  X )
107, 8, 93syl 19 . . . . . . . . . . 11  |-  ( ph  ->  -.  X  e.  X
)
11 fvex 5744 . . . . . . . . . . . . . 14  |-  ( G `
 X )  e. 
_V
12 dif1o 6746 . . . . . . . . . . . . . 14  |-  ( ( G `  X )  e.  ( _V  \  1o )  <->  ( ( G `
 X )  e. 
_V  /\  ( G `  X )  =/=  (/) ) )
1311, 12mpbiran 886 . . . . . . . . . . . . 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 7623 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
1814, 17mpbid 203 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
1918simpld 447 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G : B --> A )
20 ffn 5593 . . . . . . . . . . . . . . . 16  |-  ( G : B --> A  ->  G  Fn  B )
21 elpreima 5852 . . . . . . . . . . . . . . . 16  |-  ( G  Fn  B  ->  ( X  e.  ( `' G " ( _V  \  1o ) )  <->  ( X  e.  B  /\  ( G `  X )  e.  ( _V  \  1o ) ) ) )
2219, 20, 213syl 19 . . . . . . . . . . . . . . 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 3349 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( X  e.  ( `' G " ( _V 
\  1o ) )  ->  X  e.  X
) )
2522, 24sylbird 228 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( X  e.  B  /\  ( G `
 X )  e.  ( _V  \  1o ) )  ->  X  e.  X ) )
265, 25mpand 658 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( G `  X )  e.  ( _V  \  1o )  ->  X  e.  X
) )
2713, 26syl5bir 211 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( G `  X )  =/=  (/)  ->  X  e.  X ) )
2827necon1bd 2674 . . . . . . . . . . 11  |-  ( ph  ->  ( -.  X  e.  X  ->  ( G `  X )  =  (/) ) )
2910, 28mpd 15 . . . . . . . . . 10  |-  ( ph  ->  ( G `  X
)  =  (/) )
3029ad3antrrr 712 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  ( G `  X )  =  (/) )
31 simpr 449 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  t  =  X )
3231fveq2d 5734 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  ( G `  t )  =  ( G `  X ) )
33 simpllr 737 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  Y  =  (/) )
3430, 32, 333eqtr4rd 2481 . . . . . . . 8  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  t  =  X
)  ->  Y  =  ( G `  t ) )
35 eqidd 2439 . . . . . . . 8  |-  ( ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  /\  -.  t  =  X )  ->  ( G `  t )  =  ( G `  t ) )
362, 3, 34, 35ifbothda 3771 . . . . . . 7  |-  ( ( ( ph  /\  Y  =  (/) )  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  ( G `
 t ) )
3736mpteq2dva 4297 . . . . . 6  |-  ( (
ph  /\  Y  =  (/) )  ->  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( G `  t ) ) )  =  ( t  e.  B  |->  ( G `  t ) ) )
381, 37syl5eq 2482 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  F  =  ( t  e.  B  |->  ( G `  t
) ) )
3919feqmptd 5781 . . . . . 6  |-  ( ph  ->  G  =  ( t  e.  B  |->  ( G `
 t ) ) )
4039adantr 453 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  G  =  ( t  e.  B  |->  ( G `  t
) ) )
4138, 40eqtr4d 2473 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  F  =  G )
4214adantr 453 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  G  e.  S )
4341, 42eqeltrd 2512 . . 3  |-  ( (
ph  /\  Y  =  (/) )  ->  F  e.  S )
44 oecl 6783 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
4516, 4, 44syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( A  ^o  B
)  e.  On )
4615, 16, 4cantnff 7631 . . . . . . . 8  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
4746, 14ffvelrnd 5873 . . . . . . 7  |-  ( ph  ->  ( ( A CNF  B
) `  G )  e.  ( A  ^o  B
) )
48 onelon 4608 . . . . . . 7  |-  ( ( ( A  ^o  B
)  e.  On  /\  ( ( A CNF  B
) `  G )  e.  ( A  ^o  B
) )  ->  (
( A CNF  B ) `
 G )  e.  On )
4945, 47, 48syl2anc 644 . . . . . 6  |-  ( ph  ->  ( ( A CNF  B
) `  G )  e.  On )
5049adantr 453 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  G
)  e.  On )
51 oa0r 6784 . . . . 5  |-  ( ( ( A CNF  B ) `
 G )  e.  On  ->  ( (/)  +o  (
( A CNF  B ) `
 G ) )  =  ( ( A CNF 
B ) `  G
) )
5250, 51syl 16 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( (/)  +o  (
( A CNF  B ) `
 G ) )  =  ( ( A CNF 
B ) `  G
) )
53 oveq2 6091 . . . . . 6  |-  ( Y  =  (/)  ->  ( ( A  ^o  X )  .o  Y )  =  ( ( A  ^o  X )  .o  (/) ) )
54 oecl 6783 . . . . . . . 8  |-  ( ( A  e.  On  /\  X  e.  On )  ->  ( A  ^o  X
)  e.  On )
5516, 7, 54syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( A  ^o  X
)  e.  On )
56 om0 6763 . . . . . . 7  |-  ( ( A  ^o  X )  e.  On  ->  (
( A  ^o  X
)  .o  (/) )  =  (/) )
5755, 56syl 16 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  X )  .o  (/) )  =  (/) )
5853, 57sylan9eqr 2492 . . . . 5  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A  ^o  X )  .o  Y )  =  (/) )
5958oveq1d 6098 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( (
( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) )  =  (
(/)  +o  ( ( A CNF  B ) `  G
) ) )
6041fveq2d 5734 . . . 4  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( A CNF  B ) `  G ) )
6152, 59, 603eqtr4rd 2481 . . 3  |-  ( (
ph  /\  Y  =  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( ( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) ) )
6243, 61jca 520 . 2  |-  ( (
ph  /\  Y  =  (/) )  ->  ( F  e.  S  /\  (
( A CNF  B ) `
 F )  =  ( ( ( A  ^o  X )  .o  Y )  +o  (
( A CNF  B ) `
 G ) ) ) )
6316adantr 453 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  A  e.  On )
644adantr 453 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  B  e.  On )
6514adantr 453 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  G  e.  S )
665adantr 453 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  X  e.  B )
67 cantnfp1.6 . . . . 5  |-  ( ph  ->  Y  e.  A )
6867adantr 453 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  Y  e.  A )
6923adantr 453 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  ( `' G " ( _V  \  1o ) )  C_  X
)
7015, 63, 64, 65, 66, 68, 69, 1cantnfp1lem1 7636 . . 3  |-  ( (
ph  /\  Y  =/=  (/) )  ->  F  e.  S )
71 onelon 4608 . . . . . . 7  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
7216, 67, 71syl2anc 644 . . . . . 6  |-  ( ph  ->  Y  e.  On )
73 on0eln0 4638 . . . . . 6  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
7472, 73syl 16 . . . . 5  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
7574biimpar 473 . . . 4  |-  ( (
ph  /\  Y  =/=  (/) )  ->  (/)  e.  Y
)
76 eqid 2438 . . . 4  |- OrdIso (  _E  ,  ( `' F " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' F " ( _V 
\  1o ) ) )
77 eqid 2438 . . . 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
) ) ,  (/) )
78 eqid 2438 . . . 4  |- OrdIso (  _E  ,  ( `' G " ( _V  \  1o ) ) )  = OrdIso
(  _E  ,  ( `' G " ( _V 
\  1o ) ) )
79 eqid 2438 . . . 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
) ) ,  (/) )
8015, 63, 64, 65, 66, 68, 69, 1, 75, 76, 77, 78, 79cantnfp1lem3 7638 . . 3  |-  ( (
ph  /\  Y  =/=  (/) )  ->  ( ( A CNF  B ) `  F
)  =  ( ( ( A  ^o  X
)  .o  Y )  +o  ( ( A CNF 
B ) `  G
) ) )
8170, 80jca 520 . 2  |-  ( (
ph  /\  Y  =/=  (/) )  ->  ( F  e.  S  /\  (
( A CNF  B ) `
 F )  =  ( ( ( A  ^o  X )  .o  Y )  +o  (
( A CNF  B ) `
 G ) ) ) )
8262, 81pm2.61dane 2684 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958    \ cdif 3319    C_ wss 3322   (/)c0 3630   ifcif 3741    e. cmpt 4268    _E cep 4494   Ord word 4582   Oncon0 4583   `'ccnv 4879   dom cdm 4880   "cima 4883    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085  seq𝜔cseqom 6706   1oc1o 6719    +o coa 6723    .o comu 6724    ^o coe 6725   Fincfn 7111  OrdIsocoi 7480   CNF ccnf 7618
This theorem is referenced by:  cantnflem1d  7646  cantnflem1  7647  cantnflem3  7649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-seqom 6707  df-1o 6726  df-2o 6727  df-oadd 6730  df-omul 6731  df-oexp 6732  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-oi 7481  df-cnf 7619
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