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Theorem fseq1p1m1 9058
Description: Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 7-Mar-2014.)
Hypothesis
Ref Expression
fseq1p1m1.1  |-  H  =  { <. ( N  + 
1 ) ,  B >. }
Assertion
Ref Expression
fseq1p1m1  |-  ( N  e.  NN0  ->  ( ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
)  <->  ( G :
( 1 ... ( N  +  1 ) ) --> A  /\  ( G `  ( N  +  1 ) )  =  B  /\  F  =  ( G  |`  ( 1 ... N
) ) ) ) )

Proof of Theorem fseq1p1m1
StepHypRef Expression
1 simpr1 921 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  F : ( 1 ... N ) --> A )
2 nn0p1nn 8278 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
32adantr 265 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( N  +  1 )  e.  NN )
4 simpr2 922 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  B  e.  A )
5 fseq1p1m1.1 . . . . . . . . 9  |-  H  =  { <. ( N  + 
1 ) ,  B >. }
6 fsng 5364 . . . . . . . . 9  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  ( H : {
( N  +  1 ) } --> { B } 
<->  H  =  { <. ( N  +  1 ) ,  B >. } ) )
75, 6mpbiri 161 . . . . . . . 8  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  H : { ( N  +  1 ) } --> { B }
)
83, 4, 7syl2anc 397 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  H : { ( N  + 
1 ) } --> { B } )
94snssd 3537 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  { B }  C_  A )
108, 9fssd 5083 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  H : { ( N  + 
1 ) } --> A )
11 fzp1disj 9044 . . . . . . 7  |-  ( ( 1 ... N )  i^i  { ( N  +  1 ) } )  =  (/)
1211a1i 9 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( 1 ... N
)  i^i  { ( N  +  1 ) } )  =  (/) )
13 fun2 5092 . . . . . 6  |-  ( ( ( F : ( 1 ... N ) --> A  /\  H : { ( N  + 
1 ) } --> A )  /\  ( ( 1 ... N )  i^i 
{ ( N  + 
1 ) } )  =  (/) )  ->  ( F  u.  H ) : ( ( 1 ... N )  u. 
{ ( N  + 
1 ) } ) --> A )
141, 10, 12, 13syl21anc 1145 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( F  u.  H ) : ( ( 1 ... N )  u. 
{ ( N  + 
1 ) } ) --> A )
15 1z 8328 . . . . . . . 8  |-  1  e.  ZZ
16 simpl 106 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  N  e.  NN0 )
17 nn0uz 8603 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
18 1m1e0 8059 . . . . . . . . . . 11  |-  ( 1  -  1 )  =  0
1918fveq2i 5209 . . . . . . . . . 10  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
2017, 19eqtr4i 2079 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
2116, 20syl6eleq 2146 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  N  e.  ( ZZ>= `  ( 1  -  1 ) ) )
22 fzsuc2 9043 . . . . . . . 8  |-  ( ( 1  e.  ZZ  /\  N  e.  ( ZZ>= `  ( 1  -  1 ) ) )  -> 
( 1 ... ( N  +  1 ) )  =  ( ( 1 ... N )  u.  { ( N  +  1 ) } ) )
2315, 21, 22sylancr 399 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
1 ... ( N  + 
1 ) )  =  ( ( 1 ... N )  u.  {
( N  +  1 ) } ) )
2423eqcomd 2061 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( 1 ... N
)  u.  { ( N  +  1 ) } )  =  ( 1 ... ( N  +  1 ) ) )
2524feq2d 5063 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( F  u.  H
) : ( ( 1 ... N )  u.  { ( N  +  1 ) } ) --> A  <->  ( F  u.  H ) : ( 1 ... ( N  +  1 ) ) --> A ) )
2614, 25mpbid 139 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( F  u.  H ) : ( 1 ... ( N  +  1 ) ) --> A )
27 simpr3 923 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  G  =  ( F  u.  H ) )
2827feq1d 5062 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( G : ( 1 ... ( N  +  1 ) ) --> A  <->  ( F  u.  H ) : ( 1 ... ( N  +  1 ) ) --> A ) )
2926, 28mpbird 160 . . 3  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  G : ( 1 ... ( N  +  1 ) ) --> A )
3027reseq1d 4639 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( G  |`  { ( N  +  1 ) } )  =  ( ( F  u.  H )  |`  { ( N  + 
1 ) } ) )
31 ffn 5074 . . . . . . . . . 10  |-  ( F : ( 1 ... N ) --> A  ->  F  Fn  ( 1 ... N ) )
32 fnresdisj 5037 . . . . . . . . . 10  |-  ( F  Fn  ( 1 ... N )  ->  (
( ( 1 ... N )  i^i  {
( N  +  1 ) } )  =  (/) 
<->  ( F  |`  { ( N  +  1 ) } )  =  (/) ) )
331, 31, 323syl 17 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( ( 1 ... N )  i^i  {
( N  +  1 ) } )  =  (/) 
<->  ( F  |`  { ( N  +  1 ) } )  =  (/) ) )
3412, 33mpbid 139 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( F  |`  { ( N  +  1 ) } )  =  (/) )
3534uneq1d 3124 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( F  |`  { ( N  +  1 ) } )  u.  ( H  |`  { ( N  +  1 ) } ) )  =  (
(/)  u.  ( H  |` 
{ ( N  + 
1 ) } ) ) )
36 resundir 4654 . . . . . . 7  |-  ( ( F  u.  H )  |`  { ( N  + 
1 ) } )  =  ( ( F  |`  { ( N  + 
1 ) } )  u.  ( H  |`  { ( N  + 
1 ) } ) )
37 uncom 3115 . . . . . . . 8  |-  ( (/)  u.  ( H  |`  { ( N  +  1 ) } ) )  =  ( ( H  |`  { ( N  + 
1 ) } )  u.  (/) )
38 un0 3279 . . . . . . . 8  |-  ( ( H  |`  { ( N  +  1 ) } )  u.  (/) )  =  ( H  |`  { ( N  +  1 ) } )
3937, 38eqtr2i 2077 . . . . . . 7  |-  ( H  |`  { ( N  + 
1 ) } )  =  ( (/)  u.  ( H  |`  { ( N  +  1 ) } ) )
4035, 36, 393eqtr4g 2113 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( F  u.  H
)  |`  { ( N  +  1 ) } )  =  ( H  |`  { ( N  + 
1 ) } ) )
41 ffn 5074 . . . . . . 7  |-  ( H : { ( N  +  1 ) } --> A  ->  H  Fn  { ( N  +  1 ) } )
42 fnresdm 5036 . . . . . . 7  |-  ( H  Fn  { ( N  +  1 ) }  ->  ( H  |`  { ( N  + 
1 ) } )  =  H )
4310, 41, 423syl 17 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( H  |`  { ( N  +  1 ) } )  =  H )
4430, 40, 433eqtrd 2092 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( G  |`  { ( N  +  1 ) } )  =  H )
4544fveq1d 5208 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( G  |`  { ( N  +  1 ) } ) `  ( N  +  1 ) )  =  ( H `
 ( N  + 
1 ) ) )
4616nn0zd 8417 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  N  e.  ZZ )
4746peano2zd 8422 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( N  +  1 )  e.  ZZ )
48 snidg 3428 . . . . 5  |-  ( ( N  +  1 )  e.  ZZ  ->  ( N  +  1 )  e.  { ( N  +  1 ) } )
49 fvres 5226 . . . . 5  |-  ( ( N  +  1 )  e.  { ( N  +  1 ) }  ->  ( ( G  |`  { ( N  + 
1 ) } ) `
 ( N  + 
1 ) )  =  ( G `  ( N  +  1 ) ) )
5047, 48, 493syl 17 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( G  |`  { ( N  +  1 ) } ) `  ( N  +  1 ) )  =  ( G `
 ( N  + 
1 ) ) )
515fveq1i 5207 . . . . . 6  |-  ( H `
 ( N  + 
1 ) )  =  ( { <. ( N  +  1 ) ,  B >. } `  ( N  +  1
) )
52 fvsng 5387 . . . . . 6  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  ( { <. ( N  +  1 ) ,  B >. } `  ( N  +  1
) )  =  B )
5351, 52syl5eq 2100 . . . . 5  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  ( H `  ( N  +  1 ) )  =  B )
543, 4, 53syl2anc 397 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( H `  ( N  +  1 ) )  =  B )
5545, 50, 543eqtr3d 2096 . . 3  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( G `  ( N  +  1 ) )  =  B )
5627reseq1d 4639 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( G  |`  ( 1 ... N ) )  =  ( ( F  u.  H )  |`  (
1 ... N ) ) )
57 incom 3157 . . . . . . . 8  |-  ( { ( N  +  1 ) }  i^i  (
1 ... N ) )  =  ( ( 1 ... N )  i^i 
{ ( N  + 
1 ) } )
5857, 12syl5eq 2100 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( { ( N  + 
1 ) }  i^i  ( 1 ... N
) )  =  (/) )
59 ffn 5074 . . . . . . . 8  |-  ( H : { ( N  +  1 ) } --> { B }  ->  H  Fn  { ( N  +  1 ) } )
60 fnresdisj 5037 . . . . . . . 8  |-  ( H  Fn  { ( N  +  1 ) }  ->  ( ( { ( N  +  1 ) }  i^i  (
1 ... N ) )  =  (/)  <->  ( H  |`  ( 1 ... N
) )  =  (/) ) )
618, 59, 603syl 17 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( { ( N  +  1 ) }  i^i  ( 1 ... N ) )  =  (/) 
<->  ( H  |`  (
1 ... N ) )  =  (/) ) )
6258, 61mpbid 139 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( H  |`  ( 1 ... N ) )  =  (/) )
6362uneq2d 3125 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( F  |`  (
1 ... N ) )  u.  ( H  |`  ( 1 ... N
) ) )  =  ( ( F  |`  ( 1 ... N
) )  u.  (/) ) )
64 resundir 4654 . . . . 5  |-  ( ( F  u.  H )  |`  ( 1 ... N
) )  =  ( ( F  |`  (
1 ... N ) )  u.  ( H  |`  ( 1 ... N
) ) )
65 un0 3279 . . . . . 6  |-  ( ( F  |`  ( 1 ... N ) )  u.  (/) )  =  ( F  |`  ( 1 ... N ) )
6665eqcomi 2060 . . . . 5  |-  ( F  |`  ( 1 ... N
) )  =  ( ( F  |`  (
1 ... N ) )  u.  (/) )
6763, 64, 663eqtr4g 2113 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( F  u.  H
)  |`  ( 1 ... N ) )  =  ( F  |`  (
1 ... N ) ) )
68 fnresdm 5036 . . . . 5  |-  ( F  Fn  ( 1 ... N )  ->  ( F  |`  ( 1 ... N ) )  =  F )
691, 31, 683syl 17 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( F  |`  ( 1 ... N ) )  =  F )
7056, 67, 693eqtrrd 2093 . . 3  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  F  =  ( G  |`  ( 1 ... N
) ) )
7129, 55, 703jca 1095 . 2  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( G : ( 1 ... ( N  +  1 ) ) --> A  /\  ( G `  ( N  +  1 ) )  =  B  /\  F  =  ( G  |`  ( 1 ... N
) ) ) )
72 simpr1 921 . . . . 5  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  G : ( 1 ... ( N  +  1 ) ) --> A )
73 fzssp1 9032 . . . . 5  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
74 fssres 5094 . . . . 5  |-  ( ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( 1 ... N )  C_  (
1 ... ( N  + 
1 ) ) )  ->  ( G  |`  ( 1 ... N
) ) : ( 1 ... N ) --> A )
7572, 73, 74sylancl 398 . . . 4  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G  |`  ( 1 ... N ) ) : ( 1 ... N
) --> A )
76 simpr3 923 . . . . 5  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  F  =  ( G  |`  ( 1 ... N
) ) )
7776feq1d 5062 . . . 4  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( F : ( 1 ... N ) --> A  <->  ( G  |`  ( 1 ... N
) ) : ( 1 ... N ) --> A ) )
7875, 77mpbird 160 . . 3  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  F : ( 1 ... N ) --> A )
79 simpr2 922 . . . 4  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G `  ( N  +  1 ) )  =  B )
802adantr 265 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( N  +  1 )  e.  NN )
81 nnuz 8604 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
8280, 81syl6eleq 2146 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( N  +  1 )  e.  ( ZZ>= `  1
) )
83 eluzfz2 8998 . . . . . 6  |-  ( ( N  +  1 )  e.  ( ZZ>= `  1
)  ->  ( N  +  1 )  e.  ( 1 ... ( N  +  1 ) ) )
8482, 83syl 14 . . . . 5  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( N  +  1 )  e.  ( 1 ... ( N  +  1 ) ) )
8572, 84ffvelrnd 5331 . . . 4  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G `  ( N  +  1 ) )  e.  A )
8679, 85eqeltrrd 2131 . . 3  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  B  e.  A )
87 ffn 5074 . . . . . . . . 9  |-  ( G : ( 1 ... ( N  +  1 ) ) --> A  ->  G  Fn  ( 1 ... ( N  + 
1 ) ) )
8872, 87syl 14 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  G  Fn  ( 1 ... ( N  +  1 ) ) )
89 fnressn 5377 . . . . . . . 8  |-  ( ( G  Fn  ( 1 ... ( N  + 
1 ) )  /\  ( N  +  1
)  e.  ( 1 ... ( N  + 
1 ) ) )  ->  ( G  |`  { ( N  + 
1 ) } )  =  { <. ( N  +  1 ) ,  ( G `  ( N  +  1
) ) >. } )
9088, 84, 89syl2anc 397 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G  |`  { ( N  +  1 ) } )  =  { <. ( N  +  1 ) ,  ( G `  ( N  +  1
) ) >. } )
91 opeq2 3578 . . . . . . . . 9  |-  ( ( G `  ( N  +  1 ) )  =  B  ->  <. ( N  +  1 ) ,  ( G `  ( N  +  1
) ) >.  =  <. ( N  +  1 ) ,  B >. )
9291sneqd 3416 . . . . . . . 8  |-  ( ( G `  ( N  +  1 ) )  =  B  ->  { <. ( N  +  1 ) ,  ( G `  ( N  +  1
) ) >. }  =  { <. ( N  + 
1 ) ,  B >. } )
9379, 92syl 14 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  { <. ( N  +  1 ) ,  ( G `  ( N  +  1
) ) >. }  =  { <. ( N  + 
1 ) ,  B >. } )
9490, 93eqtrd 2088 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G  |`  { ( N  +  1 ) } )  =  { <. ( N  +  1 ) ,  B >. } )
9594, 5syl6reqr 2107 . . . . 5  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  H  =  ( G  |`  { ( N  + 
1 ) } ) )
9676, 95uneq12d 3126 . . . 4  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( F  u.  H )  =  ( ( G  |`  ( 1 ... N
) )  u.  ( G  |`  { ( N  +  1 ) } ) ) )
97 simpl 106 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  N  e.  NN0 )
9897, 20syl6eleq 2146 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  N  e.  ( ZZ>= `  ( 1  -  1 ) ) )
9915, 98, 22sylancr 399 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  (
1 ... ( N  + 
1 ) )  =  ( ( 1 ... N )  u.  {
( N  +  1 ) } ) )
10099reseq2d 4640 . . . . 5  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G  |`  ( 1 ... ( N  +  1 ) ) )  =  ( G  |`  (
( 1 ... N
)  u.  { ( N  +  1 ) } ) ) )
101 resundi 4653 . . . . 5  |-  ( G  |`  ( ( 1 ... N )  u.  {
( N  +  1 ) } ) )  =  ( ( G  |`  ( 1 ... N
) )  u.  ( G  |`  { ( N  +  1 ) } ) )
102100, 101syl6req 2105 . . . 4  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  (
( G  |`  (
1 ... N ) )  u.  ( G  |`  { ( N  + 
1 ) } ) )  =  ( G  |`  ( 1 ... ( N  +  1 ) ) ) )
103 fnresdm 5036 . . . . 5  |-  ( G  Fn  ( 1 ... ( N  +  1 ) )  ->  ( G  |`  ( 1 ... ( N  +  1 ) ) )  =  G )
10472, 87, 1033syl 17 . . . 4  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G  |`  ( 1 ... ( N  +  1 ) ) )  =  G )
10596, 102, 1043eqtrrd 2093 . . 3  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  G  =  ( F  u.  H ) )
10678, 86, 1053jca 1095 . 2  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H ) ) )
10771, 106impbida 538 1  |-  ( N  e.  NN0  ->  ( ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
)  <->  ( G :
( 1 ... ( N  +  1 ) ) --> A  /\  ( G `  ( N  +  1 ) )  =  B  /\  F  =  ( G  |`  ( 1 ... N
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409    u. cun 2943    i^i cin 2944    C_ wss 2945   (/)c0 3252   {csn 3403   <.cop 3406    |` cres 4375    Fn wfn 4925   -->wf 4926   ` cfv 4930  (class class class)co 5540   0cc0 6947   1c1 6948    + caddc 6950    - cmin 7245   NNcn 7990   NN0cn0 8239   ZZcz 8302   ZZ>=cuz 8569   ...cfz 8976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-inn 7991  df-n0 8240  df-z 8303  df-uz 8570  df-fz 8977
This theorem is referenced by:  fseq1m1p1  9059
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