Theorem fseq1p1m1 10163

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

Step	Hyp	Ref	Expression
1		simpr1 1005	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> F : ( 1 ... N ) --> A )
2		nn0p1nn 9282	. . . . . . . . 9 |- ( N e. NN0 -> ( N + 1 ) e. NN )
3	2	adantr 276	. . . . . . . 8 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( N + 1 ) e. NN )
4		simpr2 1006	. . . . . . . 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 5732	. . . . . . . . 9 |- ( ( ( N + 1 ) e. NN /\ B e. A ) -> ( H : { ( N + 1 ) } --> { B } <-> H = { <. ( N + 1 ) , B >. } ) )
7	5, 6	mpbiri 168	. . . . . . . 8 |- ( ( ( N + 1 ) e. NN /\ B e. A ) -> H : { ( N + 1 ) } --> { B } )
8	3, 4, 7	syl2anc 411	. . . . . . 7 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> H : { ( N + 1 ) } --> { B } )
9	4	snssd 3764	. . . . . . 7 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> { B } C_ A )
10	8, 9	fssd 5417	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> H : { ( N + 1 ) } --> A )
11		fzp1disj 10149	. . . . . . 7 |- ( ( 1 ... N ) i^i { ( N + 1 ) } ) = (/)
12	11	a1i 9	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( ( 1 ... N ) i^i { ( N + 1 ) } ) = (/) )
13		fun2 5428	. . . . . 6 |- ( ( ( F : ( 1 ... N ) --> A /\ H : { ( N + 1 ) } --> A ) /\ ( ( 1 ... N ) i^i { ( N + 1 ) } ) = (/) ) -> ( F u. H ) : ( ( 1 ... N ) u. { ( N + 1 ) } ) --> A )
14	1, 10, 12, 13	syl21anc 1248	. . . . 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 9346	. . . . . . . 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 9630	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
18		1m1e0 9053	. . . . . . . . . . 11 |- ( 1 - 1 ) = 0
19	18	fveq2i 5558	. . . . . . . . . 10 |- ( ZZ>= ` ( 1 - 1 ) ) = ( ZZ>= ` 0 )
20	17, 19	eqtr4i 2217	. . . . . . . . 9 |- NN0 = ( ZZ>= ` ( 1 - 1 ) )
21	16, 20	eleqtrdi 2286	. . . . . . . 8 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> N e. ( ZZ>= ` ( 1 - 1 ) ) )
22		fzsuc2 10148	. . . . . . . 8 |- ( ( 1 e. ZZ /\ N e. ( ZZ>= ` ( 1 - 1 ) ) ) -> ( 1 ... ( N + 1 ) ) = ( ( 1 ... N ) u. { ( N + 1 ) } ) )
23	15, 21, 22	sylancr 414	. . . . . . 7 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( 1 ... ( N + 1 ) ) = ( ( 1 ... N ) u. { ( N + 1 ) } ) )
24	23	eqcomd 2199	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( ( 1 ... N ) u. { ( N + 1 ) } ) = ( 1 ... ( N + 1 ) ) )
25	24	feq2d 5392	. . . . 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 ) )
26	14, 25	mpbid 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 1007	. . . . 5 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> G = ( F u. H ) )
28	27	feq1d 5391	. . . 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 ) )
29	26, 28	mpbird 167	. . 3 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> G : ( 1 ... ( N + 1 ) ) --> A )
30	27	reseq1d 4942	. . . . . 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 5404	. . . . . . . . . 10 |- ( F : ( 1 ... N ) --> A -> F Fn ( 1 ... N ) )
32		fnresdisj 5365	. . . . . . . . . 10 |- ( F Fn ( 1 ... N ) -> ( ( ( 1 ... N ) i^i { ( N + 1 ) } ) = (/) <-> ( F |` { ( N + 1 ) } ) = (/) ) )
33	1, 31, 32	3syl 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 ) } ) = (/) ) )
34	12, 33	mpbid 147	. . . . . . . 8 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( F |` { ( N + 1 ) } ) = (/) )
35	34	uneq1d 3313	. . . . . . 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 4957	. . . . . . 7 |- ( ( F u. H ) |` { ( N + 1 ) } ) = ( ( F |` { ( N + 1 ) } ) u. ( H |` { ( N + 1 ) } ) )
37		uncom 3304	. . . . . . . 8 |- ( (/) u. ( H |` { ( N + 1 ) } ) ) = ( ( H |` { ( N + 1 ) } ) u. (/) )
38		un0 3481	. . . . . . . 8 |- ( ( H |` { ( N + 1 ) } ) u. (/) ) = ( H |` { ( N + 1 ) } )
39	37, 38	eqtr2i 2215	. . . . . . 7 |- ( H |` { ( N + 1 ) } ) = ( (/) u. ( H |` { ( N + 1 ) } ) )
40	35, 36, 39	3eqtr4g 2251	. . . . . 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 5404	. . . . . . 7 |- ( H : { ( N + 1 ) } --> A -> H Fn { ( N + 1 ) } )
42		fnresdm 5364	. . . . . . 7 |- ( H Fn { ( N + 1 ) } -> ( H |` { ( N + 1 ) } ) = H )
43	10, 41, 42	3syl 17	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( H |` { ( N + 1 ) } ) = H )
44	30, 40, 43	3eqtrd 2230	. . . . 5 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( G |` { ( N + 1 ) } ) = H )
45	44	fveq1d 5557	. . . 4 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( ( G |` { ( N + 1 ) } ) ` ( N + 1 ) ) = ( H ` ( N + 1 ) ) )
46	16	nn0zd 9440	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> N e. ZZ )
47	46	peano2zd 9445	. . . . 5 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( N + 1 ) e. ZZ )
48		snidg 3648	. . . . 5 |- ( ( N + 1 ) e. ZZ -> ( N + 1 ) e. { ( N + 1 ) } )
49		fvres 5579	. . . . 5 |- ( ( N + 1 ) e. { ( N + 1 ) } -> ( ( G |` { ( N + 1 ) } ) ` ( N + 1 ) ) = ( G ` ( N + 1 ) ) )
50	47, 48, 49	3syl 17	. . . 4 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( ( G |` { ( N + 1 ) } ) ` ( N + 1 ) ) = ( G ` ( N + 1 ) ) )
51	5	fveq1i 5556	. . . . . 6 |- ( H ` ( N + 1 ) ) = ( { <. ( N + 1 ) , B >. } ` ( N + 1 ) )
52		fvsng 5755	. . . . . 6 |- ( ( ( N + 1 ) e. NN /\ B e. A ) -> ( { <. ( N + 1 ) , B >. } ` ( N + 1 ) ) = B )
53	51, 52	eqtrid 2238	. . . . 5 |- ( ( ( N + 1 ) e. NN /\ B e. A ) -> ( H ` ( N + 1 ) ) = B )
54	3, 4, 53	syl2anc 411	. . . 4 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( H ` ( N + 1 ) ) = B )
55	45, 50, 54	3eqtr3d 2234	. . 3 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( G ` ( N + 1 ) ) = B )
56	27	reseq1d 4942	. . . 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 3352	. . . . . . . 8 |- ( { ( N + 1 ) } i^i ( 1 ... N ) ) = ( ( 1 ... N ) i^i { ( N + 1 ) } )
58	57, 12	eqtrid 2238	. . . . . . 7 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( { ( N + 1 ) } i^i ( 1 ... N ) ) = (/) )
59		ffn 5404	. . . . . . . 8 |- ( H : { ( N + 1 ) } --> { B } -> H Fn { ( N + 1 ) } )
60		fnresdisj 5365	. . . . . . . 8 |- ( H Fn { ( N + 1 ) } -> ( ( { ( N + 1 ) } i^i ( 1 ... N ) ) = (/) <-> ( H |` ( 1 ... N ) ) = (/) ) )
61	8, 59, 60	3syl 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 ) ) = (/) ) )
62	58, 61	mpbid 147	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( H |` ( 1 ... N ) ) = (/) )
63	62	uneq2d 3314	. . . . 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 4957	. . . . 5 |- ( ( F u. H ) |` ( 1 ... N ) ) = ( ( F |` ( 1 ... N ) ) u. ( H |` ( 1 ... N ) ) )
65		un0 3481	. . . . . 6 |- ( ( F |` ( 1 ... N ) ) u. (/) ) = ( F |` ( 1 ... N ) )
66	65	eqcomi 2197	. . . . 5 |- ( F |` ( 1 ... N ) ) = ( ( F |` ( 1 ... N ) ) u. (/) )
67	63, 64, 66	3eqtr4g 2251	. . . 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 5364	. . . . 5 |- ( F Fn ( 1 ... N ) -> ( F |` ( 1 ... N ) ) = F )
69	1, 31, 68	3syl 17	. . . 4 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( F |` ( 1 ... N ) ) = F )
70	56, 67, 69	3eqtrrd 2231	. . 3 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> F = ( G |` ( 1 ... N ) ) )
71	29, 55, 70	3jca 1179	. 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 1005	. . . . 5 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> G : ( 1 ... ( N + 1 ) ) --> A )
73		fzssp1 10136	. . . . 5 |- ( 1 ... N ) C_ ( 1 ... ( N + 1 ) )
74		fssres 5430	. . . . 5 |- ( ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( 1 ... N ) C_ ( 1 ... ( N + 1 ) ) ) -> ( G |` ( 1 ... N ) ) : ( 1 ... N ) --> A )
75	72, 73, 74	sylancl 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 1007	. . . . 5 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> F = ( G |` ( 1 ... N ) ) )
77	76	feq1d 5391	. . . 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 ) )
78	75, 77	mpbird 167	. . 3 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> F : ( 1 ... N ) --> A )
79		simpr2 1006	. . . 4 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> ( G ` ( N + 1 ) ) = B )
80	2	adantr 276	. . . . . . 7 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> ( N + 1 ) e. NN )
81		nnuz 9631	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
82	80, 81	eleqtrdi 2286	. . . . . 6 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> ( N + 1 ) e. ( ZZ>= ` 1 ) )
83		eluzfz2 10101	. . . . . 6 |- ( ( N + 1 ) e. ( ZZ>= ` 1 ) -> ( N + 1 ) e. ( 1 ... ( N + 1 ) ) )
84	82, 83	syl 14	. . . . 5 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> ( N + 1 ) e. ( 1 ... ( N + 1 ) ) )
85	72, 84	ffvelcdmd 5695	. . . 4 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> ( G ` ( N + 1 ) ) e. A )
86	79, 85	eqeltrrd 2271	. . 3 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> B e. A )
87		ffn 5404	. . . . . . . . 9 |- ( G : ( 1 ... ( N + 1 ) ) --> A -> G Fn ( 1 ... ( N + 1 ) ) )
88	72, 87	syl 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 5745	. . . . . . . 8 |- ( ( G Fn ( 1 ... ( N + 1 ) ) /\ ( N + 1 ) e. ( 1 ... ( N + 1 ) ) ) -> ( G |` { ( N + 1 ) } ) = { <. ( N + 1 ) , ( G ` ( N + 1 ) ) >. } )
90	88, 84, 89	syl2anc 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 3806	. . . . . . . . 9 |- ( ( G ` ( N + 1 ) ) = B -> <. ( N + 1 ) , ( G ` ( N + 1 ) ) >. = <. ( N + 1 ) , B >. )
92	91	sneqd 3632	. . . . . . . 8 |- ( ( G ` ( N + 1 ) ) = B -> { <. ( N + 1 ) , ( G ` ( N + 1 ) ) >. } = { <. ( N + 1 ) , B >. } )
93	79, 92	syl 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 >. } )
94	90, 93	eqtrd 2226	. . . . . 6 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> ( G |` { ( N + 1 ) } ) = { <. ( N + 1 ) , B >. } )
95	5, 94	eqtr4id 2245	. . . . 5 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> H = ( G |` { ( N + 1 ) } ) )
96	76, 95	uneq12d 3315	. . . 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 )
98	97, 20	eleqtrdi 2286	. . . . . . 7 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> N e. ( ZZ>= ` ( 1 - 1 ) ) )
99	15, 98, 22	sylancr 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 ) } ) )
100	99	reseq2d 4943	. . . . 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 4956	. . . . 5 |- ( G |` ( ( 1 ... N ) u. { ( N + 1 ) } ) ) = ( ( G |` ( 1 ... N ) ) u. ( G |` { ( N + 1 ) } ) )
102	100, 101	eqtr2di 2243	. . . 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 5364	. . . . 5 |- ( G Fn ( 1 ... ( N + 1 ) ) -> ( G |` ( 1 ... ( N + 1 ) ) ) = G )
104	72, 87, 103	3syl 17	. . . 4 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> ( G |` ( 1 ... ( N + 1 ) ) ) = G )
105	96, 102, 104	3eqtrrd 2231	. . 3 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> G = ( F u. H ) )
106	78, 86, 105	3jca 1179	. 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 ) ) )
107	71, 106	impbida 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 ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   u. cun 3152   i^i cin 3153   C_ wss 3154   (/)c0 3447   {csn 3619   <.cop 3622   |` cres 4662   Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5919   0cc0 7874   1c1 7875   + caddc 7877   - cmin 8192   NNcn 8984   NN0cn0 9243   ZZcz 9320   ZZ>=cuz 9595   ...cfz 10077

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-fz 10078

This theorem is referenced by:  fseq1m1p1  10164