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Theorem fseq1p1m1 11530
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 989 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  F : ( 1 ... N ) --> A )
2 nn0p1nn 10615 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
32adantr 462 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( N  +  1 )  e.  NN )
4 simpr2 990 . . . . . . . 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 5879 . . . . . . . . 9  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  ( H : {
( N  +  1 ) } --> { B } 
<->  H  =  { <. ( N  +  1 ) ,  B >. } ) )
75, 6mpbiri 233 . . . . . . . 8  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  H : { ( N  +  1 ) } --> { B }
)
83, 4, 7syl2anc 656 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  H : { ( N  + 
1 ) } --> { B } )
94snssd 4015 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  { B }  C_  A )
10 fss 5564 . . . . . . 7  |-  ( ( H : { ( N  +  1 ) } --> { B }  /\  { B }  C_  A )  ->  H : { ( N  + 
1 ) } --> A )
118, 9, 10syl2anc 656 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  H : { ( N  + 
1 ) } --> A )
12 fzp1disj 11511 . . . . . . 7  |-  ( ( 1 ... N )  i^i  { ( N  +  1 ) } )  =  (/)
1312a1i 11 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( 1 ... N
)  i^i  { ( N  +  1 ) } )  =  (/) )
14 fun2 5573 . . . . . 6  |-  ( ( ( F : ( 1 ... N ) --> A  /\  H : { ( N  + 
1 ) } --> A )  /\  ( ( 1 ... N )  i^i 
{ ( N  + 
1 ) } )  =  (/) )  ->  ( F  u.  H ) : ( ( 1 ... N )  u. 
{ ( N  + 
1 ) } ) --> A )
151, 11, 13, 14syl21anc 1212 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( F  u.  H ) : ( ( 1 ... N )  u. 
{ ( N  + 
1 ) } ) --> A )
16 1z 10672 . . . . . . . 8  |-  1  e.  ZZ
17 simpl 454 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  N  e.  NN0 )
18 nn0uz 10891 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
19 1m1e0 10386 . . . . . . . . . . 11  |-  ( 1  -  1 )  =  0
2019fveq2i 5691 . . . . . . . . . 10  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
2118, 20eqtr4i 2464 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
2217, 21syl6eleq 2531 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  N  e.  ( ZZ>= `  ( 1  -  1 ) ) )
23 fzsuc2 11510 . . . . . . . 8  |-  ( ( 1  e.  ZZ  /\  N  e.  ( ZZ>= `  ( 1  -  1 ) ) )  -> 
( 1 ... ( N  +  1 ) )  =  ( ( 1 ... N )  u.  { ( N  +  1 ) } ) )
2416, 22, 23sylancr 658 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
1 ... ( N  + 
1 ) )  =  ( ( 1 ... N )  u.  {
( N  +  1 ) } ) )
2524eqcomd 2446 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( 1 ... N
)  u.  { ( N  +  1 ) } )  =  ( 1 ... ( N  +  1 ) ) )
2625feq2d 5544 . . . . 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 ) )
2715, 26mpbid 210 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( F  u.  H ) : ( 1 ... ( N  +  1 ) ) --> A )
28 simpr3 991 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  G  =  ( F  u.  H ) )
2928feq1d 5543 . . . 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 ) )
3027, 29mpbird 232 . . 3  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  G : ( 1 ... ( N  +  1 ) ) --> A )
31 ovex 6115 . . . . . 6  |-  ( N  +  1 )  e. 
_V
3231snid 3902 . . . . 5  |-  ( N  +  1 )  e. 
{ ( N  + 
1 ) }
33 fvres 5701 . . . . 5  |-  ( ( N  +  1 )  e.  { ( N  +  1 ) }  ->  ( ( G  |`  { ( N  + 
1 ) } ) `
 ( N  + 
1 ) )  =  ( G `  ( N  +  1 ) ) )
3432, 33ax-mp 5 . . . 4  |-  ( ( G  |`  { ( N  +  1 ) } ) `  ( N  +  1 ) )  =  ( G `
 ( N  + 
1 ) )
3528reseq1d 5105 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( G  |`  { ( N  +  1 ) } )  =  ( ( F  u.  H )  |`  { ( N  + 
1 ) } ) )
36 ffn 5556 . . . . . . . . . . 11  |-  ( F : ( 1 ... N ) --> A  ->  F  Fn  ( 1 ... N ) )
37 fnresdisj 5518 . . . . . . . . . . 11  |-  ( F  Fn  ( 1 ... N )  ->  (
( ( 1 ... N )  i^i  {
( N  +  1 ) } )  =  (/) 
<->  ( F  |`  { ( N  +  1 ) } )  =  (/) ) )
381, 36, 373syl 20 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( ( 1 ... N )  i^i  {
( N  +  1 ) } )  =  (/) 
<->  ( F  |`  { ( N  +  1 ) } )  =  (/) ) )
3913, 38mpbid 210 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( F  |`  { ( N  +  1 ) } )  =  (/) )
4039uneq1d 3506 . . . . . . . 8  |-  ( ( 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 ) } ) ) )
41 resundir 5122 . . . . . . . 8  |-  ( ( F  u.  H )  |`  { ( N  + 
1 ) } )  =  ( ( F  |`  { ( N  + 
1 ) } )  u.  ( H  |`  { ( N  + 
1 ) } ) )
42 uncom 3497 . . . . . . . . 9  |-  ( (/)  u.  ( H  |`  { ( N  +  1 ) } ) )  =  ( ( H  |`  { ( N  + 
1 ) } )  u.  (/) )
43 un0 3659 . . . . . . . . 9  |-  ( ( H  |`  { ( N  +  1 ) } )  u.  (/) )  =  ( H  |`  { ( N  +  1 ) } )
4442, 43eqtr2i 2462 . . . . . . . 8  |-  ( H  |`  { ( N  + 
1 ) } )  =  ( (/)  u.  ( H  |`  { ( N  +  1 ) } ) )
4540, 41, 443eqtr4g 2498 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( F  u.  H
)  |`  { ( N  +  1 ) } )  =  ( H  |`  { ( N  + 
1 ) } ) )
46 ffn 5556 . . . . . . . 8  |-  ( H : { ( N  +  1 ) } --> A  ->  H  Fn  { ( N  +  1 ) } )
47 fnresdm 5517 . . . . . . . 8  |-  ( H  Fn  { ( N  +  1 ) }  ->  ( H  |`  { ( N  + 
1 ) } )  =  H )
4811, 46, 473syl 20 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( H  |`  { ( N  +  1 ) } )  =  H )
4935, 45, 483eqtrd 2477 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( G  |`  { ( N  +  1 ) } )  =  H )
5049fveq1d 5690 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( G  |`  { ( N  +  1 ) } ) `  ( N  +  1 ) )  =  ( H `
 ( N  + 
1 ) ) )
515fveq1i 5689 . . . . . . 7  |-  ( H `
 ( N  + 
1 ) )  =  ( { <. ( N  +  1 ) ,  B >. } `  ( N  +  1
) )
52 fvsng 5909 . . . . . . 7  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  ( { <. ( N  +  1 ) ,  B >. } `  ( N  +  1
) )  =  B )
5351, 52syl5eq 2485 . . . . . 6  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  ( H `  ( N  +  1 ) )  =  B )
543, 4, 53syl2anc 656 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( H `  ( N  +  1 ) )  =  B )
5550, 54eqtrd 2473 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( G  |`  { ( N  +  1 ) } ) `  ( N  +  1 ) )  =  B )
5634, 55syl5eqr 2487 . . 3  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( G `  ( N  +  1 ) )  =  B )
5728reseq1d 5105 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( G  |`  ( 1 ... N ) )  =  ( ( F  u.  H )  |`  (
1 ... N ) ) )
58 incom 3540 . . . . . . . 8  |-  ( { ( N  +  1 ) }  i^i  (
1 ... N ) )  =  ( ( 1 ... N )  i^i 
{ ( N  + 
1 ) } )
5958, 13syl5eq 2485 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( { ( N  + 
1 ) }  i^i  ( 1 ... N
) )  =  (/) )
60 ffn 5556 . . . . . . . 8  |-  ( H : { ( N  +  1 ) } --> { B }  ->  H  Fn  { ( N  +  1 ) } )
61 fnresdisj 5518 . . . . . . . 8  |-  ( H  Fn  { ( N  +  1 ) }  ->  ( ( { ( N  +  1 ) }  i^i  (
1 ... N ) )  =  (/)  <->  ( H  |`  ( 1 ... N
) )  =  (/) ) )
628, 60, 613syl 20 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( { ( N  +  1 ) }  i^i  ( 1 ... N ) )  =  (/) 
<->  ( H  |`  (
1 ... N ) )  =  (/) ) )
6359, 62mpbid 210 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( H  |`  ( 1 ... N ) )  =  (/) )
6463uneq2d 3507 . . . . 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.  (/) ) )
65 resundir 5122 . . . . 5  |-  ( ( F  u.  H )  |`  ( 1 ... N
) )  =  ( ( F  |`  (
1 ... N ) )  u.  ( H  |`  ( 1 ... N
) ) )
66 un0 3659 . . . . . 6  |-  ( ( F  |`  ( 1 ... N ) )  u.  (/) )  =  ( F  |`  ( 1 ... N ) )
6766eqcomi 2445 . . . . 5  |-  ( F  |`  ( 1 ... N
) )  =  ( ( F  |`  (
1 ... N ) )  u.  (/) )
6864, 65, 673eqtr4g 2498 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( F  u.  H
)  |`  ( 1 ... N ) )  =  ( F  |`  (
1 ... N ) ) )
69 fnresdm 5517 . . . . 5  |-  ( F  Fn  ( 1 ... N )  ->  ( F  |`  ( 1 ... N ) )  =  F )
701, 36, 693syl 20 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( F  |`  ( 1 ... N ) )  =  F )
7157, 68, 703eqtrrd 2478 . . 3  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  F  =  ( G  |`  ( 1 ... N
) ) )
7230, 56, 713jca 1163 . 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
) ) ) )
73 simpr1 989 . . . . 5  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  G : ( 1 ... ( N  +  1 ) ) --> A )
74 fzssp1 11497 . . . . 5  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
75 fssres 5575 . . . . 5  |-  ( ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( 1 ... N )  C_  (
1 ... ( N  + 
1 ) ) )  ->  ( G  |`  ( 1 ... N
) ) : ( 1 ... N ) --> A )
7673, 74, 75sylancl 657 . . . 4  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G  |`  ( 1 ... N ) ) : ( 1 ... N
) --> A )
77 simpr3 991 . . . . 5  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  F  =  ( G  |`  ( 1 ... N
) ) )
7877feq1d 5543 . . . 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 ) )
7976, 78mpbird 232 . . 3  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  F : ( 1 ... N ) --> A )
80 simpr2 990 . . . 4  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G `  ( N  +  1 ) )  =  B )
812adantr 462 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( N  +  1 )  e.  NN )
82 nnuz 10892 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
8381, 82syl6eleq 2531 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( N  +  1 )  e.  ( ZZ>= `  1
) )
84 eluzfz2 11455 . . . . . 6  |-  ( ( N  +  1 )  e.  ( ZZ>= `  1
)  ->  ( N  +  1 )  e.  ( 1 ... ( N  +  1 ) ) )
8583, 84syl 16 . . . . 5  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( N  +  1 )  e.  ( 1 ... ( N  +  1 ) ) )
8673, 85ffvelrnd 5841 . . . 4  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G `  ( N  +  1 ) )  e.  A )
8780, 86eqeltrrd 2516 . . 3  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  B  e.  A )
88 ffn 5556 . . . . . . . . 9  |-  ( G : ( 1 ... ( N  +  1 ) ) --> A  ->  G  Fn  ( 1 ... ( N  + 
1 ) ) )
8973, 88syl 16 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  G  Fn  ( 1 ... ( N  +  1 ) ) )
90 fnressn 5891 . . . . . . . 8  |-  ( ( G  Fn  ( 1 ... ( N  + 
1 ) )  /\  ( N  +  1
)  e.  ( 1 ... ( N  + 
1 ) ) )  ->  ( G  |`  { ( N  + 
1 ) } )  =  { <. ( N  +  1 ) ,  ( G `  ( N  +  1
) ) >. } )
9189, 85, 90syl2anc 656 . . . . . . 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
) ) >. } )
92 opeq2 4057 . . . . . . . . 9  |-  ( ( G `  ( N  +  1 ) )  =  B  ->  <. ( N  +  1 ) ,  ( G `  ( N  +  1
) ) >.  =  <. ( N  +  1 ) ,  B >. )
9392sneqd 3886 . . . . . . . 8  |-  ( ( G `  ( N  +  1 ) )  =  B  ->  { <. ( N  +  1 ) ,  ( G `  ( N  +  1
) ) >. }  =  { <. ( N  + 
1 ) ,  B >. } )
9480, 93syl 16 . . . . . . 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 >. } )
9591, 94eqtrd 2473 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G  |`  { ( N  +  1 ) } )  =  { <. ( N  +  1 ) ,  B >. } )
9695, 5syl6reqr 2492 . . . . 5  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  H  =  ( G  |`  { ( N  + 
1 ) } ) )
9777, 96uneq12d 3508 . . . 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 ) } ) ) )
98 simpl 454 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  N  e.  NN0 )
9998, 21syl6eleq 2531 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  N  e.  ( ZZ>= `  ( 1  -  1 ) ) )
10016, 99, 23sylancr 658 . . . . . 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 ) } ) )
101100reseq2d 5106 . . . . 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 ) } ) ) )
102 resundi 5121 . . . . 5  |-  ( G  |`  ( ( 1 ... N )  u.  {
( N  +  1 ) } ) )  =  ( ( G  |`  ( 1 ... N
) )  u.  ( G  |`  { ( N  +  1 ) } ) )
103101, 102syl6req 2490 . . . 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 ) ) ) )
104 fnresdm 5517 . . . . 5  |-  ( G  Fn  ( 1 ... ( N  +  1 ) )  ->  ( G  |`  ( 1 ... ( N  +  1 ) ) )  =  G )
10573, 88, 1043syl 20 . . . 4  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G  |`  ( 1 ... ( N  +  1 ) ) )  =  G )
10697, 103, 1053eqtrrd 2478 . . 3  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  G  =  ( F  u.  H ) )
10779, 87, 1063jca 1163 . 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 ) ) )
10872, 107impbida 823 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    u. cun 3323    i^i cin 3324    C_ wss 3325   (/)c0 3634   {csn 3874   <.cop 3880    |` cres 4838    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   0cc0 9278   1c1 9279    + caddc 9281    - cmin 9591   NNcn 10318   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434
This theorem is referenced by:  fseq1m1p1  11531
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