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Theorem fseq1p1m1 10091
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 1003 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  F : ( 1 ... N ) --> A )
2 nn0p1nn 9213 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
32adantr 276 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( N  +  1 )  e.  NN )
4 simpr2 1004 . . . . . . . 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 5689 . . . . . . . . 9  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  ( H : {
( N  +  1 ) } --> { B } 
<->  H  =  { <. ( N  +  1 ) ,  B >. } ) )
75, 6mpbiri 168 . . . . . . . 8  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  H : { ( N  +  1 ) } --> { B }
)
83, 4, 7syl2anc 411 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  H : { ( N  + 
1 ) } --> { B } )
94snssd 3737 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  { B }  C_  A )
108, 9fssd 5378 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  H : { ( N  + 
1 ) } --> A )
11 fzp1disj 10077 . . . . . . 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 5389 . . . . . 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 1237 . . . . 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 9277 . . . . . . . 8  |-  1  e.  ZZ
16 simpl 109 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  N  e.  NN0 )
17 nn0uz 9560 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
18 1m1e0 8986 . . . . . . . . . . 11  |-  ( 1  -  1 )  =  0
1918fveq2i 5518 . . . . . . . . . 10  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
2017, 19eqtr4i 2201 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
2116, 20eleqtrdi 2270 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  N  e.  ( ZZ>= `  ( 1  -  1 ) ) )
22 fzsuc2 10076 . . . . . . . 8  |-  ( ( 1  e.  ZZ  /\  N  e.  ( ZZ>= `  ( 1  -  1 ) ) )  -> 
( 1 ... ( N  +  1 ) )  =  ( ( 1 ... N )  u.  { ( N  +  1 ) } ) )
2315, 21, 22sylancr 414 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
1 ... ( N  + 
1 ) )  =  ( ( 1 ... N )  u.  {
( N  +  1 ) } ) )
2423eqcomd 2183 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( 1 ... N
)  u.  { ( N  +  1 ) } )  =  ( 1 ... ( N  +  1 ) ) )
2524feq2d 5353 . . . . 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 147 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( F  u.  H ) : ( 1 ... ( N  +  1 ) ) --> A )
27 simpr3 1005 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  G  =  ( F  u.  H ) )
2827feq1d 5352 . . . 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 167 . . 3  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  G : ( 1 ... ( N  +  1 ) ) --> A )
3027reseq1d 4906 . . . . . 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 5365 . . . . . . . . . 10  |-  ( F : ( 1 ... N ) --> A  ->  F  Fn  ( 1 ... N ) )
32 fnresdisj 5326 . . . . . . . . . 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 147 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( F  |`  { ( N  +  1 ) } )  =  (/) )
3534uneq1d 3288 . . . . . . 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 4921 . . . . . . 7  |-  ( ( F  u.  H )  |`  { ( N  + 
1 ) } )  =  ( ( F  |`  { ( N  + 
1 ) } )  u.  ( H  |`  { ( N  + 
1 ) } ) )
37 uncom 3279 . . . . . . . 8  |-  ( (/)  u.  ( H  |`  { ( N  +  1 ) } ) )  =  ( ( H  |`  { ( N  + 
1 ) } )  u.  (/) )
38 un0 3456 . . . . . . . 8  |-  ( ( H  |`  { ( N  +  1 ) } )  u.  (/) )  =  ( H  |`  { ( N  +  1 ) } )
3937, 38eqtr2i 2199 . . . . . . 7  |-  ( H  |`  { ( N  + 
1 ) } )  =  ( (/)  u.  ( H  |`  { ( N  +  1 ) } ) )
4035, 36, 393eqtr4g 2235 . . . . . 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 5365 . . . . . . 7  |-  ( H : { ( N  +  1 ) } --> A  ->  H  Fn  { ( N  +  1 ) } )
42 fnresdm 5325 . . . . . . 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 2214 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( G  |`  { ( N  +  1 ) } )  =  H )
4544fveq1d 5517 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  (
( G  |`  { ( N  +  1 ) } ) `  ( N  +  1 ) )  =  ( H `
 ( N  + 
1 ) ) )
4616nn0zd 9371 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  N  e.  ZZ )
4746peano2zd 9376 . . . . 5  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( N  +  1 )  e.  ZZ )
48 snidg 3621 . . . . 5  |-  ( ( N  +  1 )  e.  ZZ  ->  ( N  +  1 )  e.  { ( N  +  1 ) } )
49 fvres 5539 . . . . 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 5516 . . . . . 6  |-  ( H `
 ( N  + 
1 ) )  =  ( { <. ( N  +  1 ) ,  B >. } `  ( N  +  1
) )
52 fvsng 5712 . . . . . 6  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  ( { <. ( N  +  1 ) ,  B >. } `  ( N  +  1
) )  =  B )
5351, 52eqtrid 2222 . . . . 5  |-  ( ( ( N  +  1 )  e.  NN  /\  B  e.  A )  ->  ( H `  ( N  +  1 ) )  =  B )
543, 4, 53syl2anc 411 . . . 4  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( H `  ( N  +  1 ) )  =  B )
5545, 50, 543eqtr3d 2218 . . 3  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( G `  ( N  +  1 ) )  =  B )
5627reseq1d 4906 . . . 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 3327 . . . . . . . 8  |-  ( { ( N  +  1 ) }  i^i  (
1 ... N ) )  =  ( ( 1 ... N )  i^i 
{ ( N  + 
1 ) } )
5857, 12eqtrid 2222 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( { ( N  + 
1 ) }  i^i  ( 1 ... N
) )  =  (/) )
59 ffn 5365 . . . . . . . 8  |-  ( H : { ( N  +  1 ) } --> { B }  ->  H  Fn  { ( N  +  1 ) } )
60 fnresdisj 5326 . . . . . . . 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 147 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  ( H  |`  ( 1 ... N ) )  =  (/) )
6362uneq2d 3289 . . . . 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 4921 . . . . 5  |-  ( ( F  u.  H )  |`  ( 1 ... N
) )  =  ( ( F  |`  (
1 ... N ) )  u.  ( H  |`  ( 1 ... N
) ) )
65 un0 3456 . . . . . 6  |-  ( ( F  |`  ( 1 ... N ) )  u.  (/) )  =  ( F  |`  ( 1 ... N ) )
6665eqcomi 2181 . . . . 5  |-  ( F  |`  ( 1 ... N
) )  =  ( ( F  |`  (
1 ... N ) )  u.  (/) )
6763, 64, 663eqtr4g 2235 . . . 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 5325 . . . . 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 2215 . . 3  |-  ( ( N  e.  NN0  /\  ( F : ( 1 ... N ) --> A  /\  B  e.  A  /\  G  =  ( F  u.  H )
) )  ->  F  =  ( G  |`  ( 1 ... N
) ) )
7129, 55, 703jca 1177 . 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 1003 . . . . 5  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  G : ( 1 ... ( N  +  1 ) ) --> A )
73 fzssp1 10064 . . . . 5  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
74 fssres 5391 . . . . 5  |-  ( ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( 1 ... N )  C_  (
1 ... ( N  + 
1 ) ) )  ->  ( G  |`  ( 1 ... N
) ) : ( 1 ... N ) --> A )
7572, 73, 74sylancl 413 . . . 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 1005 . . . . 5  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  F  =  ( G  |`  ( 1 ... N
) ) )
7776feq1d 5352 . . . 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 167 . . 3  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  F : ( 1 ... N ) --> A )
79 simpr2 1004 . . . 4  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G `  ( N  +  1 ) )  =  B )
802adantr 276 . . . . . . 7  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( N  +  1 )  e.  NN )
81 nnuz 9561 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
8280, 81eleqtrdi 2270 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( N  +  1 )  e.  ( ZZ>= `  1
) )
83 eluzfz2 10029 . . . . . 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, 84ffvelcdmd 5652 . . . 4  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G `  ( N  +  1 ) )  e.  A )
8679, 85eqeltrrd 2255 . . 3  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  B  e.  A )
87 ffn 5365 . . . . . . . . 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 5702 . . . . . . . 8  |-  ( ( G  Fn  ( 1 ... ( N  + 
1 ) )  /\  ( N  +  1
)  e.  ( 1 ... ( N  + 
1 ) ) )  ->  ( G  |`  { ( N  + 
1 ) } )  =  { <. ( N  +  1 ) ,  ( G `  ( N  +  1
) ) >. } )
9088, 84, 89syl2anc 411 . . . . . . 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 3779 . . . . . . . . 9  |-  ( ( G `  ( N  +  1 ) )  =  B  ->  <. ( N  +  1 ) ,  ( G `  ( N  +  1
) ) >.  =  <. ( N  +  1 ) ,  B >. )
9291sneqd 3605 . . . . . . . 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 2210 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  ( G  |`  { ( N  +  1 ) } )  =  { <. ( N  +  1 ) ,  B >. } )
955, 94eqtr4id 2229 . . . . 5  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  H  =  ( G  |`  { ( N  + 
1 ) } ) )
9676, 95uneq12d 3290 . . . 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 109 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  N  e.  NN0 )
9897, 20eleqtrdi 2270 . . . . . . 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 414 . . . . . 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 4907 . . . . 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 4920 . . . . 5  |-  ( G  |`  ( ( 1 ... N )  u.  {
( N  +  1 ) } ) )  =  ( ( G  |`  ( 1 ... N
) )  u.  ( G  |`  { ( N  +  1 ) } ) )
102100, 101eqtr2di 2227 . . . 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 5325 . . . . 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 2215 . . 3  |-  ( ( N  e.  NN0  /\  ( G : ( 1 ... ( N  + 
1 ) ) --> A  /\  ( G `  ( N  +  1
) )  =  B  /\  F  =  ( G  |`  ( 1 ... N ) ) ) )  ->  G  =  ( F  u.  H ) )
10678, 86, 1053jca 1177 . 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 596 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 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148    u. cun 3127    i^i cin 3128    C_ wss 3129   (/)c0 3422   {csn 3592   <.cop 3595    |` cres 4628    Fn wfn 5211   -->wf 5212   ` cfv 5216  (class class class)co 5874   0cc0 7810   1c1 7811    + caddc 7813    - cmin 8126   NNcn 8917   NN0cn0 9174   ZZcz 9251   ZZ>=cuz 9526   ...cfz 10006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-inn 8918  df-n0 9175  df-z 9252  df-uz 9527  df-fz 10007
This theorem is referenced by:  fseq1m1p1  10092
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