Theorem fseq1p1m1 10251

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 1006	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> F : ( 1 ... N ) --> A )
2		nn0p1nn 9369	. . . . . . . . 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 1007	. . . . . . . 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 5776	. . . . . . . . 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 3789	. . . . . . 7 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> { B } C_ A )
10	8, 9	fssd 5458	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> H : { ( N + 1 ) } --> A )
11		fzp1disj 10237	. . . . . . 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 5470	. . . . . 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 1249	. . . . 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 9433	. . . . . . . 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 9718	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
18		1m1e0 9140	. . . . . . . . . . 11 |- ( 1 - 1 ) = 0
19	18	fveq2i 5602	. . . . . . . . . 10 |- ( ZZ>= ` ( 1 - 1 ) ) = ( ZZ>= ` 0 )
20	17, 19	eqtr4i 2231	. . . . . . . . 9 |- NN0 = ( ZZ>= ` ( 1 - 1 ) )
21	16, 20	eleqtrdi 2300	. . . . . . . 8 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> N e. ( ZZ>= ` ( 1 - 1 ) ) )
22		fzsuc2 10236	. . . . . . . 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 2213	. . . . . 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 5433	. . . . 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 1008	. . . . 5 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> G = ( F u. H ) )
28	27	feq1d 5432	. . . 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 4977	. . . . . 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 5445	. . . . . . . . . 10 |- ( F : ( 1 ... N ) --> A -> F Fn ( 1 ... N ) )
32		fnresdisj 5405	. . . . . . . . . 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 3334	. . . . . . 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 4992	. . . . . . 7 |- ( ( F u. H ) |` { ( N + 1 ) } ) = ( ( F |` { ( N + 1 ) } ) u. ( H |` { ( N + 1 ) } ) )
37		uncom 3325	. . . . . . . 8 |- ( (/) u. ( H |` { ( N + 1 ) } ) ) = ( ( H |` { ( N + 1 ) } ) u. (/) )
38		un0 3502	. . . . . . . 8 |- ( ( H |` { ( N + 1 ) } ) u. (/) ) = ( H |` { ( N + 1 ) } )
39	37, 38	eqtr2i 2229	. . . . . . 7 |- ( H |` { ( N + 1 ) } ) = ( (/) u. ( H |` { ( N + 1 ) } ) )
40	35, 36, 39	3eqtr4g 2265	. . . . . 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 5445	. . . . . . 7 |- ( H : { ( N + 1 ) } --> A -> H Fn { ( N + 1 ) } )
42		fnresdm 5404	. . . . . . 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 2244	. . . . 5 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( G |` { ( N + 1 ) } ) = H )
45	44	fveq1d 5601	. . . 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 9528	. . . . . 6 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> N e. ZZ )
47	46	peano2zd 9533	. . . . 5 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( N + 1 ) e. ZZ )
48		snidg 3672	. . . . 5 |- ( ( N + 1 ) e. ZZ -> ( N + 1 ) e. { ( N + 1 ) } )
49		fvres 5623	. . . . 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 5600	. . . . . 6 |- ( H ` ( N + 1 ) ) = ( { <. ( N + 1 ) , B >. } ` ( N + 1 ) )
52		fvsng 5803	. . . . . 6 |- ( ( ( N + 1 ) e. NN /\ B e. A ) -> ( { <. ( N + 1 ) , B >. } ` ( N + 1 ) ) = B )
53	51, 52	eqtrid 2252	. . . . 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 2248	. . 3 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( G ` ( N + 1 ) ) = B )
56	27	reseq1d 4977	. . . 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 3373	. . . . . . . 8 |- ( { ( N + 1 ) } i^i ( 1 ... N ) ) = ( ( 1 ... N ) i^i { ( N + 1 ) } )
58	57, 12	eqtrid 2252	. . . . . . 7 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> ( { ( N + 1 ) } i^i ( 1 ... N ) ) = (/) )
59		ffn 5445	. . . . . . . 8 |- ( H : { ( N + 1 ) } --> { B } -> H Fn { ( N + 1 ) } )
60		fnresdisj 5405	. . . . . . . 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 3335	. . . . 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 4992	. . . . 5 |- ( ( F u. H ) |` ( 1 ... N ) ) = ( ( F |` ( 1 ... N ) ) u. ( H |` ( 1 ... N ) ) )
65		un0 3502	. . . . . 6 |- ( ( F |` ( 1 ... N ) ) u. (/) ) = ( F |` ( 1 ... N ) )
66	65	eqcomi 2211	. . . . 5 |- ( F |` ( 1 ... N ) ) = ( ( F |` ( 1 ... N ) ) u. (/) )
67	63, 64, 66	3eqtr4g 2265	. . . 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 5404	. . . . 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 2245	. . 3 |- ( ( N e. NN0 /\ ( F : ( 1 ... N ) --> A /\ B e. A /\ G = ( F u. H ) ) ) -> F = ( G |` ( 1 ... N ) ) )
71	29, 55, 70	3jca 1180	. 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 1006	. . . . 5 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> G : ( 1 ... ( N + 1 ) ) --> A )
73		fzssp1 10224	. . . . 5 |- ( 1 ... N ) C_ ( 1 ... ( N + 1 ) )
74		fssres 5473	. . . . 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 1008	. . . . 5 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> F = ( G |` ( 1 ... N ) ) )
77	76	feq1d 5432	. . . 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 1007	. . . 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 9719	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
82	80, 81	eleqtrdi 2300	. . . . . 6 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> ( N + 1 ) e. ( ZZ>= ` 1 ) )
83		eluzfz2 10189	. . . . . 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 5739	. . . 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 2285	. . 3 |- ( ( N e. NN0 /\ ( G : ( 1 ... ( N + 1 ) ) --> A /\ ( G ` ( N + 1 ) ) = B /\ F = ( G |` ( 1 ... N ) ) ) ) -> B e. A )
87		ffn 5445	. . . . . . . . 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 5793	. . . . . . . 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 3834	. . . . . . . . 9 |- ( ( G ` ( N + 1 ) ) = B -> <. ( N + 1 ) , ( G ` ( N + 1 ) ) >. = <. ( N + 1 ) , B >. )
92	91	sneqd 3656	. . . . . . . 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 2240	. . . . . 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 2259	. . . . 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 3336	. . . 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 2300	. . . . . . 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 4978	. . . . 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 4991	. . . . 5 |- ( G |` ( ( 1 ... N ) u. { ( N + 1 ) } ) ) = ( ( G |` ( 1 ... N ) ) u. ( G |` { ( N + 1 ) } ) )
102	100, 101	eqtr2di 2257	. . . 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 5404	. . . . 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 2245	. . 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 1180	. 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 981   = wceq 1373   e. wcel 2178   u. cun 3172   i^i cin 3173   C_ wss 3174   (/)c0 3468   {csn 3643   <.cop 3646   |` cres 4695   Fn wfn 5285   -->wf 5286   ` cfv 5290  (class class class)co 5967   0cc0 7960   1c1 7961   + caddc 7963   - cmin 8278   NNcn 9071   NN0cn0 9330   ZZcz 9407   ZZ>=cuz 9683   ...cfz 10165

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166

This theorem is referenced by:  fseq1m1p1  10252