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Theorem fseq1p1m1 11077
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 963 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  F : ( 1 ... N ) --> A )
2 nn0p1nn 10215 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
32adantr 452 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( N  +  1 )  e.  NN )
4 simpr2 964 . . . . . . . 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 5866 . . . . . . . . 9  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  ( H : {
( N  +  1 ) } --> { B } 
<->  H  =  { <. ( N  +  1 ) ,  B >. } ) )
75, 6mpbiri 225 . . . . . . . 8  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  H : { ( N  +  1 ) } --> { B }
)
83, 4, 7syl2anc 643 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  H : { ( N  + 
1 ) } --> { B } )
94snssd 3903 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  { B }  C_  A )
10 fss 5558 . . . . . . 7  |-  ( ( H : { ( N  +  1 ) } --> { B }  /\  { B }  C_  A )  ->  H : { ( N  + 
1 ) } --> A )
118, 9, 10syl2anc 643 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  H : { ( N  + 
1 ) } --> A )
12 fzp1disj 11061 . . . . . . 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 5567 . . . . . 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 1183 . . . . 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 10267 . . . . . . . 8  |-  1  e.  ZZ
17 simpl 444 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  N  e.  NN0 )
18 nn0uz 10476 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
19 1m1e0 10024 . . . . . . . . . . 11  |-  ( 1  -  1 )  =  0
2019fveq2i 5690 . . . . . . . . . 10  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
2118, 20eqtr4i 2427 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
2217, 21syl6eleq 2494 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  N  e.  ( ZZ>= `  ( 1  -  1 ) ) )
23 fzsuc2 11060 . . . . . . . 8  |-  ( ( 1  e.  ZZ  /\  N  e.  ( ZZ>= `  ( 1  -  1 ) ) )  -> 
( 1 ... ( N  +  1 ) )  =  ( ( 1 ... N )  u.  { ( N  +  1 ) } ) )
2416, 22, 23sylancr 645 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
1 ... ( N  + 
1 ) )  =  ( ( 1 ... N )  u.  {
( N  +  1 ) } ) )
2524eqcomd 2409 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( 1 ... N
)  u.  { ( N  +  1 ) } )  =  ( 1 ... ( N  +  1 ) ) )
2625feq2d 5540 . . . . 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 202 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( F  u.  H ) : ( 1 ... ( N  +  1 ) ) --> A )
28 simpr3 965 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  G  =  ( F  u.  H ) )
2928feq1d 5539 . . . 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 224 . . 3  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  G : ( 1 ... ( N  +  1 ) ) --> A )
31 ovex 6065 . . . . . 6  |-  ( N  +  1 )  e. 
_V
3231snid 3801 . . . . 5  |-  ( N  +  1 )  e. 
{ ( N  + 
1 ) }
33 fvres 5704 . . . . 5  |-  ( ( N  +  1 )  e.  { ( N  +  1 ) }  ->  ( ( G  |`  { ( N  + 
1 ) } ) `
 ( N  + 
1 ) )  =  ( G `  ( N  +  1 ) ) )
3432, 33ax-mp 8 . . . 4  |-  ( ( G  |`  { ( N  +  1 ) } ) `  ( N  +  1 ) )  =  ( G `
 ( N  + 
1 ) )
3528reseq1d 5104 . . . . . . 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 5550 . . . . . . . . . . 11  |-  ( F : ( 1 ... N ) --> A  ->  F  Fn  ( 1 ... N ) )
37 fnresdisj 5514 . . . . . . . . . . 11  |-  ( F  Fn  ( 1 ... N )  ->  (
( ( 1 ... N )  i^i  {
( N  +  1 ) } )  =  (/) 
<->  ( F  |`  { ( N  +  1 ) } )  =  (/) ) )
381, 36, 373syl 19 . . . . . . . . . 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 202 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( F  |`  { ( N  +  1 ) } )  =  (/) )
4039uneq1d 3460 . . . . . . . 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 5120 . . . . . . . 8  |-  ( ( F  u.  H )  |`  { ( N  + 
1 ) } )  =  ( ( F  |`  { ( N  + 
1 ) } )  u.  ( H  |`  { ( N  + 
1 ) } ) )
42 uncom 3451 . . . . . . . . 9  |-  ( (/)  u.  ( H  |`  { ( N  +  1 ) } ) )  =  ( ( H  |`  { ( N  + 
1 ) } )  u.  (/) )
43 un0 3612 . . . . . . . . 9  |-  ( ( H  |`  { ( N  +  1 ) } )  u.  (/) )  =  ( H  |`  { ( N  +  1 ) } )
4442, 43eqtr2i 2425 . . . . . . . 8  |-  ( H  |`  { ( N  + 
1 ) } )  =  ( (/)  u.  ( H  |`  { ( N  +  1 ) } ) )
4540, 41, 443eqtr4g 2461 . . . . . . 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 5550 . . . . . . . 8  |-  ( H : { ( N  +  1 ) } --> A  ->  H  Fn  { ( N  +  1 ) } )
47 fnresdm 5513 . . . . . . . 8  |-  ( H  Fn  { ( N  +  1 ) }  ->  ( H  |`  { ( N  + 
1 ) } )  =  H )
4811, 46, 473syl 19 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( H  |`  { ( N  +  1 ) } )  =  H )
4935, 45, 483eqtrd 2440 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( G  |`  { ( N  +  1 ) } )  =  H )
5049fveq1d 5689 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( G  |`  { ( N  +  1 ) } ) `  ( N  +  1 ) )  =  ( H `
 ( N  + 
1 ) ) )
515fveq1i 5688 . . . . . . 7  |-  ( H `
 ( N  + 
1 ) )  =  ( { <. ( N  +  1 ) ,  B >. } `  ( N  +  1
) )
52 fvsng 5886 . . . . . . 7  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  ( { <. ( N  +  1 ) ,  B >. } `  ( N  +  1
) )  =  B )
5351, 52syl5eq 2448 . . . . . 6  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  ( H `  ( N  +  1 ) )  =  B )
543, 4, 53syl2anc 643 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( H `  ( N  +  1 ) )  =  B )
5550, 54eqtrd 2436 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( G  |`  { ( N  +  1 ) } ) `  ( N  +  1 ) )  =  B )
5634, 55syl5eqr 2450 . . 3  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( G `  ( N  +  1 ) )  =  B )
5728reseq1d 5104 . . . 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 3493 . . . . . . . 8  |-  ( { ( N  +  1 ) }  i^i  (
1 ... N ) )  =  ( ( 1 ... N )  i^i 
{ ( N  + 
1 ) } )
5958, 13syl5eq 2448 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( { ( N  + 
1 ) }  i^i  ( 1 ... N
) )  =  (/) )
60 ffn 5550 . . . . . . . 8  |-  ( H : { ( N  +  1 ) } --> { B }  ->  H  Fn  { ( N  +  1 ) } )
61 fnresdisj 5514 . . . . . . . 8  |-  ( H  Fn  { ( N  +  1 ) }  ->  ( ( { ( N  +  1 ) }  i^i  (
1 ... N ) )  =  (/)  <->  ( H  |`  ( 1 ... N
) )  =  (/) ) )
628, 60, 613syl 19 . . . . . . 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 202 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( H  |`  ( 1 ... N ) )  =  (/) )
6463uneq2d 3461 . . . . 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 5120 . . . . 5  |-  ( ( F  u.  H )  |`  ( 1 ... N
) )  =  ( ( F  |`  (
1 ... N ) )  u.  ( H  |`  ( 1 ... N
) ) )
66 un0 3612 . . . . . 6  |-  ( ( F  |`  ( 1 ... N ) )  u.  (/) )  =  ( F  |`  ( 1 ... N ) )
6766eqcomi 2408 . . . . 5  |-  ( F  |`  ( 1 ... N
) )  =  ( ( F  |`  (
1 ... N ) )  u.  (/) )
6864, 65, 673eqtr4g 2461 . . . 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 5513 . . . . 5  |-  ( F  Fn  ( 1 ... N )  ->  ( F  |`  ( 1 ... N ) )  =  F )
701, 36, 693syl 19 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( F  |`  ( 1 ... N ) )  =  F )
7157, 68, 703eqtrrd 2441 . . 3  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  F  =  ( G  |`  ( 1 ... N
) ) )
7230, 56, 713jca 1134 . 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 963 . . . . 5  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  G : ( 1 ... ( N  +  1 ) ) --> A )
74 fzssp1 11051 . . . . 5  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
75 fssres 5569 . . . . 5  |-  ( ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( 1 ... N )  C_  (
1 ... ( N  + 
1 ) ) )  ->  ( G  |`  ( 1 ... N
) ) : ( 1 ... N ) --> A )
7673, 74, 75sylancl 644 . . . 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 965 . . . . 5  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  F  =  ( G  |`  ( 1 ... N
) ) )
7877feq1d 5539 . . . 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 224 . . 3  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  F : ( 1 ... N ) --> A )
80 simpr2 964 . . . 4  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G `  ( N  +  1 ) )  =  B )
812adantr 452 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( N  +  1 )  e.  NN )
82 nnuz 10477 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
8381, 82syl6eleq 2494 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( N  +  1 )  e.  ( ZZ>= `  1
) )
84 eluzfz2 11021 . . . . . 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 5830 . . . 4  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G `  ( N  +  1 ) )  e.  A )
8780, 86eqeltrrd 2479 . . 3  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  B  e.  A )
88 ffn 5550 . . . . . . . . 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 5877 . . . . . . . 8  |-  ( ( G  Fn  ( 1 ... ( N  + 
1 ) )  /\  ( N  +  1
)  e.  ( 1 ... ( N  + 
1 ) ) )  ->  ( G  |`  { ( N  + 
1 ) } )  =  { <. ( N  +  1 ) ,  ( G `  ( N  +  1
) ) >. } )
9189, 85, 90syl2anc 643 . . . . . . 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 3945 . . . . . . . . 9  |-  ( ( G `  ( N  +  1 ) )  =  B  ->  <. ( N  +  1 ) ,  ( G `  ( N  +  1
) ) >.  =  <. ( N  +  1 ) ,  B >. )
9392sneqd 3787 . . . . . . . 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 2436 . . . . . 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 2455 . . . . 5  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  H  =  ( G  |`  { ( N  + 
1 ) } ) )
9777, 96uneq12d 3462 . . . 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 444 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  N  e.  NN0 )
9998, 21syl6eleq 2494 . . . . . . 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 645 . . . . . 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 5105 . . . . 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 5119 . . . . 5  |-  ( G  |`  ( ( 1 ... N )  u.  {
( N  +  1 ) } ) )  =  ( ( G  |`  ( 1 ... N
) )  u.  ( G  |`  { ( N  +  1 ) } ) )
103101, 102syl6req 2453 . . . 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 5513 . . . . 5  |-  ( G  Fn  ( 1 ... ( N  +  1 ) )  ->  ( G  |`  ( 1 ... ( N  +  1 ) ) )  =  G )
10573, 88, 1043syl 19 . . . 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 2441 . . 3  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  G  =  ( F  u.  H ) )
10779, 87, 1063jca 1134 . 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 806 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    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774   <.cop 3777    |` cres 4839    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   0cc0 8946   1c1 8947    + caddc 8949    - cmin 9247   NNcn 9956   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999
This theorem is referenced by:  fseq1m1p1  11078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000
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