Theorem fseq1p1m1 10080

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 1003	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> F : ( 1 ... N ) --> A )
2		nn0p1nn 9204	. . . . . . . . 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 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 5685	. . . . . . . . 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 3736	. . . . . . 7 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> { B } C_ A )
10	8, 9	fssd 5374	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> H : { ( N + 1 ) } --> A )
11		fzp1disj 10066	. . . . . . 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 5385	. . . . . 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 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 9268	. . . . . . . 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 9551	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
18		1m1e0 8977	. . . . . . . . . . 11 |- ( 1 - 1 ) = 0
19	18	fveq2i 5514	. . . . . . . . . 10 |- ( ZZ>= ` ( 1 - 1 ) ) = ( ZZ>= ` 0 )
20	17, 19	eqtr4i 2201	. . . . . . . . 9 |- NN0 = ( ZZ>= ` ( 1 - 1 ) )
21	16, 20	eleqtrdi 2270	. . . . . . . 8 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> N e. ( ZZ>= ` ( 1 - 1 ) ) )
22		fzsuc2 10065	. . . . . . . 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 2183	. . . . . 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 5349	. . . . 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 1005	. . . . 5 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> G = ( F u. H ) )
28	27	feq1d 5348	. . . 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 4902	. . . . . 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 5361	. . . . . . . . . 10 |- ( F : ( 1 ... N ) --> A -> F Fn ( 1 ... N ) )
32		fnresdisj 5322	. . . . . . . . . 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 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 4917	. . . . . . 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 ) } )
39	37, 38	eqtr2i 2199	. . . . . . 7 |- ( H |` { ( N + 1 ) } ) = ( (/) u. ( H |` { ( N + 1 ) } ) )
40	35, 36, 39	3eqtr4g 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 5361	. . . . . . 7 |- ( H : { ( N + 1 ) } --> A -> H Fn { ( N + 1 ) } )
42		fnresdm 5321	. . . . . . 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 2214	. . . . 5 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( G |` { ( N + 1 ) } ) = H )
45	44	fveq1d 5513	. . . 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 9362	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> N e. ZZ )
47	46	peano2zd 9367	. . . . 5 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( N + 1 ) e. ZZ )
48		snidg 3620	. . . . 5 |- ( ( N + 1 ) e. ZZ -> ( N + 1 ) e. { ( N + 1 ) } )
49		fvres 5535	. . . . 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 5512	. . . . . 6 |- ( H ` ( N + 1 ) ) = ( { <. ( N + 1 ) , B >. } ` ( N + 1 ) )
52		fvsng 5708	. . . . . 6 |- ( ( ( N + 1 ) e. NN /\ B e. A ) -> ( { <. ( N + 1 ) , B >. } ` ( N + 1 ) ) = B )
53	51, 52	eqtrid 2222	. . . . 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 2218	. . 3 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( G ` ( N + 1 ) ) = B )
56	27	reseq1d 4902	. . . 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 ) } )
58	57, 12	eqtrid 2222	. . . . . . 7 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( { ( N + 1 ) } i^i ( 1 ... N ) ) = (/) )
59		ffn 5361	. . . . . . . 8 |- ( H : { ( N + 1 ) } --> { B } -> H Fn { ( N + 1 ) } )
60		fnresdisj 5322	. . . . . . . 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 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 4917	. . . . 5 |- ( ( F u. H ) |` ( 1 ... N ) ) = ( ( F |` ( 1 ... N ) ) u. ( H |` ( 1 ... N ) ) )
65		un0 3456	. . . . . 6 |- ( ( F |` ( 1 ... N ) ) u. (/) ) = ( F |` ( 1 ... N ) )
66	65	eqcomi 2181	. . . . 5 |- ( F |` ( 1 ... N ) ) = ( ( F |` ( 1 ... N ) ) u. (/) )
67	63, 64, 66	3eqtr4g 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 5321	. . . . 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 2215	. . 3 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> F = ( G |` ( 1 ... N ) ) )
71	29, 55, 70	3jca 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 10053	. . . . 5 |- ( 1 ... N ) C_ ( 1 ... ( N + 1 ) )
74		fssres 5387	. . . . 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 1005	. . . . 5 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> F = ( G |` ( 1 ... N ) ) )
77	76	feq1d 5348	. . . 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 1004	. . . 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 9552	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
82	80, 81	eleqtrdi 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 10018	. . . . . 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 5648	. . . 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 2255	. . 3 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> B e. A )
87		ffn 5361	. . . . . . . . 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 5698	. . . . . . . 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 3777	. . . . . . . . 9 |- ( ( G ` ( N + 1 ) ) = B -> <. ( N + 1 ) , ( G ` ( N + 1 ) ) >. = <. ( N + 1 ) , B >. )
92	91	sneqd 3604	. . . . . . . 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 2210	. . . . . 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 2229	. . . . 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 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 )
98	97, 20	eleqtrdi 2270	. . . . . . 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 4903	. . . . 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 4916	. . . . 5 |- ( G |` ( ( 1 ... N ) u. { ( N + 1 ) } ) ) = ( ( G |` ( 1 ... N ) ) u. ( G |` { ( N + 1 ) } ) )
102	100, 101	eqtr2di 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 5321	. . . . 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 2215	. . 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 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 ) ) )
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 978   = wceq 1353   e. wcel 2148   u. cun 3127   i^i cin 3128   C_ wss 3129   (/)c0 3422   {csn 3591   <.cop 3594   |` cres 4625   Fn wfn 5207   -->wf 5208   ` cfv 5212  (class class class)co 5869   0cc0 7802   1c1 7803   + caddc 7805   - cmin 8118   NNcn 8908   NN0cn0 9165   ZZcz 9242   ZZ>=cuz 9517   ...cfz 9995

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 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918

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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996

This theorem is referenced by:  fseq1m1p1  10081