Theorem fseq1p1m1 9111

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 944	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> F : ( 1 ... N ) --> A )
2		nn0p1nn 8327	. . . . . . . . 9 |- ( N e. NN0 -> ( N + 1 ) e. NN )
3	2	adantr 270	. . . . . . . 8 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( N + 1 ) e. NN )
4		simpr2 945	. . . . . . . 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 5357	. . . . . . . . 9 |- ( ( ( N + 1 ) e. NN /\ B e. A ) -> ( H : { ( N + 1 ) } --> { B } <-> H = { <. ( N + 1 ) , B >. } ) )
7	5, 6	mpbiri 166	. . . . . . . 8 |- ( ( ( N + 1 ) e. NN /\ B e. A ) -> H : { ( N + 1 ) } --> { B } )
8	3, 4, 7	syl2anc 403	. . . . . . 7 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> H : { ( N + 1 ) } --> { B } )
9	4	snssd 3530	. . . . . . 7 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> { B } C_ A )
10	8, 9	fssd 5075	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> H : { ( N + 1 ) } --> A )
11		fzp1disj 9097	. . . . . . 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 5084	. . . . . 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 1168	. . . . 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 8377	. . . . . . . 8 |- 1 e. ZZ
16		simpl 107	. . . . . . . . 9 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> N e. NN0 )
17		nn0uz 8653	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
18		1m1e0 8108	. . . . . . . . . . 11 |- ( 1 - 1 ) = 0
19	18	fveq2i 5201	. . . . . . . . . 10 |- ( ZZ>= ` ( 1 - 1 ) ) = ( ZZ>= ` 0 )
20	17, 19	eqtr4i 2104	. . . . . . . . 9 |- NN0 = ( ZZ>= ` ( 1 - 1 ) )
21	16, 20	syl6eleq 2171	. . . . . . . 8 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> N e. ( ZZ>= ` ( 1 - 1 ) ) )
22		fzsuc2 9096	. . . . . . . 8 |- ( ( 1 e. ZZ /\ N e. ( ZZ>= ` ( 1 - 1 ) ) ) -> ( 1 ... ( N + 1 ) ) = ( ( 1 ... N ) u. { ( N + 1 ) } ) )
23	15, 21, 22	sylancr 405	. . . . . . 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 2086	. . . . . 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 5055	. . . . 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 145	. . . 4 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( F u. H ) : ( 1 ... ( N + 1 ) ) --> A )
27		simpr3 946	. . . . 5 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> G = ( F u. H ) )
28	27	feq1d 5054	. . . 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 165	. . 3 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> G : ( 1 ... ( N + 1 ) ) --> A )
30	27	reseq1d 4629	. . . . . 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 5066	. . . . . . . . . 10 |- ( F : ( 1 ... N ) --> A -> F Fn ( 1 ... N ) )
32		fnresdisj 5029	. . . . . . . . . 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 145	. . . . . . . 8 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( F |` { ( N + 1 ) } ) = (/) )
35	34	uneq1d 3125	. . . . . . 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 4644	. . . . . . 7 |- ( ( F u. H ) |` { ( N + 1 ) } ) = ( ( F |` { ( N + 1 ) } ) u. ( H |` { ( N + 1 ) } ) )
37		uncom 3116	. . . . . . . 8 |- ( (/) u. ( H |` { ( N + 1 ) } ) ) = ( ( H |` { ( N + 1 ) } ) u. (/) )
38		un0 3278	. . . . . . . 8 |- ( ( H |` { ( N + 1 ) } ) u. (/) ) = ( H |` { ( N + 1 ) } )
39	37, 38	eqtr2i 2102	. . . . . . 7 |- ( H |` { ( N + 1 ) } ) = ( (/) u. ( H |` { ( N + 1 ) } ) )
40	35, 36, 39	3eqtr4g 2138	. . . . . 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 5066	. . . . . . 7 |- ( H : { ( N + 1 ) } --> A -> H Fn { ( N + 1 ) } )
42		fnresdm 5028	. . . . . . 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 2117	. . . . 5 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( G |` { ( N + 1 ) } ) = H )
45	44	fveq1d 5200	. . . 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 8467	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> N e. ZZ )
47	46	peano2zd 8472	. . . . 5 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( N + 1 ) e. ZZ )
48		snidg 3423	. . . . 5 |- ( ( N + 1 ) e. ZZ -> ( N + 1 ) e. { ( N + 1 ) } )
49		fvres 5219	. . . . 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 5199	. . . . . 6 |- ( H ` ( N + 1 ) ) = ( { <. ( N + 1 ) , B >. } ` ( N + 1 ) )
52		fvsng 5380	. . . . . 6 |- ( ( ( N + 1 ) e. NN /\ B e. A ) -> ( { <. ( N + 1 ) , B >. } ` ( N + 1 ) ) = B )
53	51, 52	syl5eq 2125	. . . . 5 |- ( ( ( N + 1 ) e. NN /\ B e. A ) -> ( H ` ( N + 1 ) ) = B )
54	3, 4, 53	syl2anc 403	. . . 4 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( H ` ( N + 1 ) ) = B )
55	45, 50, 54	3eqtr3d 2121	. . 3 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( G ` ( N + 1 ) ) = B )
56	27	reseq1d 4629	. . . 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 3158	. . . . . . . 8 |- ( { ( N + 1 ) } i^i ( 1 ... N ) ) = ( ( 1 ... N ) i^i { ( N + 1 ) } )
58	57, 12	syl5eq 2125	. . . . . . 7 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( { ( N + 1 ) } i^i ( 1 ... N ) ) = (/) )
59		ffn 5066	. . . . . . . 8 |- ( H : { ( N + 1 ) } --> { B } -> H Fn { ( N + 1 ) } )
60		fnresdisj 5029	. . . . . . . 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 145	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( H |` ( 1 ... N ) ) = (/) )
63	62	uneq2d 3126	. . . . 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 4644	. . . . 5 |- ( ( F u. H ) |` ( 1 ... N ) ) = ( ( F |` ( 1 ... N ) ) u. ( H |` ( 1 ... N ) ) )
65		un0 3278	. . . . . 6 |- ( ( F |` ( 1 ... N ) ) u. (/) ) = ( F |` ( 1 ... N ) )
66	65	eqcomi 2085	. . . . 5 |- ( F |` ( 1 ... N ) ) = ( ( F |` ( 1 ... N ) ) u. (/) )
67	63, 64, 66	3eqtr4g 2138	. . . 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 5028	. . . . 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 2118	. . 3 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> F = ( G |` ( 1 ... N ) ) )
71	29, 55, 70	3jca 1118	. 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 944	. . . . 5 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> G : ( 1 ... ( N + 1 ) ) --> A )
73		fzssp1 9085	. . . . 5 |- ( 1 ... N ) C_ ( 1 ... ( N + 1 ) )
74		fssres 5086	. . . . 5 |- ( ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( 1 ... N ) C_ ( 1 ... ( N + 1 ) ) ) -> ( G |` ( 1 ... N ) ) : ( 1 ... N ) --> A )
75	72, 73, 74	sylancl 404	. . . 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 946	. . . . 5 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> F = ( G |` ( 1 ... N ) ) )
77	76	feq1d 5054	. . . 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 165	. . 3 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> F : ( 1 ... N ) --> A )
79		simpr2 945	. . . 4 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> ( G ` ( N + 1 ) ) = B )
80	2	adantr 270	. . . . . . 7 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> ( N + 1 ) e. NN )
81		nnuz 8654	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
82	80, 81	syl6eleq 2171	. . . . . 6 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> ( N + 1 ) e. ( ZZ>= ` 1 ) )
83		eluzfz2 9051	. . . . . 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	ffvelrnd 5324	. . . 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 2156	. . 3 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> B e. A )
87		ffn 5066	. . . . . . . . 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 5370	. . . . . . . 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 403	. . . . . . 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 3571	. . . . . . . . 9 |- ( ( G ` ( N + 1 ) ) = B -> <. ( N + 1 ) , ( G ` ( N + 1 ) ) >. = <. ( N + 1 ) , B >. )
92	91	sneqd 3411	. . . . . . . 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 2113	. . . . . 6 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> ( G |` { ( N + 1 ) } ) = { <. ( N + 1 ) , B >. } )
95	94, 5	syl6reqr 2132	. . . . 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 3127	. . . 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 107	. . . . . . . 8 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> N e. NN0 )
98	97, 20	syl6eleq 2171	. . . . . . 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 405	. . . . . 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 4630	. . . . 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 4643	. . . . 5 |- ( G |` ( ( 1 ... N ) u. { ( N + 1 ) } ) ) = ( ( G |` ( 1 ... N ) ) u. ( G |` { ( N + 1 ) } ) )
102	100, 101	syl6req 2130	. . . 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 5028	. . . . 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 2118	. . 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 1118	. 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 560	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 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   u. cun 2971   i^i cin 2972   C_ wss 2973   (/)c0 3251   {csn 3398   <.cop 3401   |` cres 4365   Fn wfn 4917   -->wf 4918   ` cfv 4922  (class class class)co 5532   0cc0 6981   1c1 6982   + caddc 6984   - cmin 7279   NNcn 8039   NN0cn0 8288   ZZcz 8351   ZZ>=cuz 8619   ...cfz 9029

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-fz 9030

This theorem is referenced by:  fseq1m1p1  9112