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 ∈ ℕ0 → ((𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 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 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐹:(1...𝑁)⟶𝐴)
2		nn0p1nn 8327	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
3	2	adantr 270	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝑁 + 1) ∈ ℕ)
4		simpr2 945	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐵 ∈ 𝐴)
5		fseq1p1m1.1	. . . . . . . . 9 ⊢ 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}
6		fsng 5357	. . . . . . . . 9 ⊢ (((𝑁 + 1) ∈ ℕ ∧ 𝐵 ∈ 𝐴) → (𝐻:{(𝑁 + 1)}⟶{𝐵} ↔ 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}))
7	5, 6	mpbiri 166	. . . . . . . 8 ⊢ (((𝑁 + 1) ∈ ℕ ∧ 𝐵 ∈ 𝐴) → 𝐻:{(𝑁 + 1)}⟶{𝐵})
8	3, 4, 7	syl2anc 403	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐻:{(𝑁 + 1)}⟶{𝐵})
9	4	snssd 3530	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → {𝐵} ⊆ 𝐴)
10	8, 9	fssd 5075	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐻:{(𝑁 + 1)}⟶𝐴)
11		fzp1disj 9097	. . . . . . 7 ⊢ ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅
12	11	a1i 9	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅)
13		fun2 5084	. . . . . 6 ⊢ (((𝐹:(1...𝑁)⟶𝐴 ∧ 𝐻:{(𝑁 + 1)}⟶𝐴) ∧ ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅) → (𝐹 ∪ 𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴)
14	1, 10, 12, 13	syl21anc 1168	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐹 ∪ 𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴)
15		1z 8377	. . . . . . . 8 ⊢ 1 ∈ ℤ
16		simpl 107	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝑁 ∈ ℕ0)
17		nn0uz 8653	. . . . . . . . . 10 ⊢ ℕ0 = (ℤ≥‘0)
18		1m1e0 8108	. . . . . . . . . . 11 ⊢ (1 − 1) = 0
19	18	fveq2i 5201	. . . . . . . . . 10 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0)
20	17, 19	eqtr4i 2104	. . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘(1 − 1))
21	16, 20	syl6eleq 2171	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝑁 ∈ (ℤ≥‘(1 − 1)))
22		fzsuc2 9096	. . . . . . . 8 ⊢ ((1 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(1 − 1))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
23	15, 21, 22	sylancr 405	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
24	23	eqcomd 2086	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((1...𝑁) ∪ {(𝑁 + 1)}) = (1...(𝑁 + 1)))
25	24	feq2d 5055	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐹 ∪ 𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴 ↔ (𝐹 ∪ 𝐻):(1...(𝑁 + 1))⟶𝐴))
26	14, 25	mpbid 145	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐹 ∪ 𝐻):(1...(𝑁 + 1))⟶𝐴)
27		simpr3 946	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐺 = (𝐹 ∪ 𝐻))
28	27	feq1d 5054	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺:(1...(𝑁 + 1))⟶𝐴 ↔ (𝐹 ∪ 𝐻):(1...(𝑁 + 1))⟶𝐴))
29	26, 28	mpbird 165	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
30	27	reseq1d 4629	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = ((𝐹 ∪ 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) ∩ {(𝑁 + 1)}) = ∅ ↔ (𝐹 ↾ {(𝑁 + 1)}) = ∅))
33	1, 31, 32	3syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ ↔ (𝐹 ↾ {(𝑁 + 1)}) = ∅))
34	12, 33	mpbid 145	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐹 ↾ {(𝑁 + 1)}) = ∅)
35	34	uneq1d 3125	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)})) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})))
36		resundir 4644	. . . . . . 7 ⊢ ((𝐹 ∪ 𝐻) ↾ {(𝑁 + 1)}) = ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)}))
37		uncom 3116	. . . . . . . 8 ⊢ (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})) = ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅)
38		un0 3278	. . . . . . . 8 ⊢ ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅) = (𝐻 ↾ {(𝑁 + 1)})
39	37, 38	eqtr2i 2102	. . . . . . 7 ⊢ (𝐻 ↾ {(𝑁 + 1)}) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)}))
40	35, 36, 39	3eqtr4g 2138	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐹 ∪ 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 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐻 ↾ {(𝑁 + 1)}) = 𝐻)
44	30, 40, 43	3eqtrd 2117	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = 𝐻)
45	44	fveq1d 5200	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐻‘(𝑁 + 1)))
46	16	nn0zd 8467	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝑁 ∈ ℤ)
47	46	peano2zd 8472	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝑁 + 1) ∈ ℤ)
48		snidg 3423	. . . . 5 ⊢ ((𝑁 + 1) ∈ ℤ → (𝑁 + 1) ∈ {(𝑁 + 1)})
49		fvres 5219	. . . . 5 ⊢ ((𝑁 + 1) ∈ {(𝑁 + 1)} → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐺‘(𝑁 + 1)))
50	47, 48, 49	3syl 17	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 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) ∈ ℕ ∧ 𝐵 ∈ 𝐴) → ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1)) = 𝐵)
53	51, 52	syl5eq 2125	. . . . 5 ⊢ (((𝑁 + 1) ∈ ℕ ∧ 𝐵 ∈ 𝐴) → (𝐻‘(𝑁 + 1)) = 𝐵)
54	3, 4, 53	syl2anc 403	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐻‘(𝑁 + 1)) = 𝐵)
55	45, 50, 54	3eqtr3d 2121	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺‘(𝑁 + 1)) = 𝐵)
56	27	reseq1d 4629	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺 ↾ (1...𝑁)) = ((𝐹 ∪ 𝐻) ↾ (1...𝑁)))
57		incom 3158	. . . . . . . 8 ⊢ ({(𝑁 + 1)} ∩ (1...𝑁)) = ((1...𝑁) ∩ {(𝑁 + 1)})
58	57, 12	syl5eq 2125	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ({(𝑁 + 1)} ∩ (1...𝑁)) = ∅)
59		ffn 5066	. . . . . . . 8 ⊢ (𝐻:{(𝑁 + 1)}⟶{𝐵} → 𝐻 Fn {(𝑁 + 1)})
60		fnresdisj 5029	. . . . . . . 8 ⊢ (𝐻 Fn {(𝑁 + 1)} → (({(𝑁 + 1)} ∩ (1...𝑁)) = ∅ ↔ (𝐻 ↾ (1...𝑁)) = ∅))
61	8, 59, 60	3syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (({(𝑁 + 1)} ∩ (1...𝑁)) = ∅ ↔ (𝐻 ↾ (1...𝑁)) = ∅))
62	58, 61	mpbid 145	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐻 ↾ (1...𝑁)) = ∅)
63	62	uneq2d 3126	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁))) = ((𝐹 ↾ (1...𝑁)) ∪ ∅))
64		resundir 4644	. . . . 5 ⊢ ((𝐹 ∪ 𝐻) ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁)))
65		un0 3278	. . . . . 6 ⊢ ((𝐹 ↾ (1...𝑁)) ∪ ∅) = (𝐹 ↾ (1...𝑁))
66	65	eqcomi 2085	. . . . 5 ⊢ (𝐹 ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ ∅)
67	63, 64, 66	3eqtr4g 2138	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐹 ∪ 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 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐹 ↾ (1...𝑁)) = 𝐹)
70	56, 67, 69	3eqtrrd 2118	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
71	29, 55, 70	3jca 1118	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁))))
72		simpr1 944	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
73		fzssp1 9085	. . . . 5 ⊢ (1...𝑁) ⊆ (1...(𝑁 + 1))
74		fssres 5086	. . . . 5 ⊢ ((𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (1...𝑁) ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶𝐴)
75	72, 73, 74	sylancl 404	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶𝐴)
76		simpr3 946	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
77	76	feq1d 5054	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (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 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹:(1...𝑁)⟶𝐴)
79		simpr2 945	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) = 𝐵)
80	2	adantr 270	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ ℕ)
81		nnuz 8654	. . . . . . 7 ⊢ ℕ = (ℤ≥‘1)
82	80, 81	syl6eleq 2171	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ (ℤ≥‘1))
83		eluzfz2 9051	. . . . . 6 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘1) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
84	82, 83	syl 14	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
85	72, 84	ffvelrnd 5324	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) ∈ 𝐴)
86	79, 85	eqeltrrd 2156	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐵 ∈ 𝐴)
87		ffn 5066	. . . . . . . . 9 ⊢ (𝐺:(1...(𝑁 + 1))⟶𝐴 → 𝐺 Fn (1...(𝑁 + 1)))
88	72, 87	syl 14	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (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) ∈ (1...(𝑁 + 1))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩})
90	88, 84, 89	syl2anc 403	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (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 ∈ ℕ0 ∧ (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 ∈ ℕ0 ∧ (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 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐻 = (𝐺 ↾ {(𝑁 + 1)}))
96	76, 95	uneq12d 3127	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹 ∪ 𝐻) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)})))
97		simpl 107	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 ∈ ℕ0)
98	97, 20	syl6eleq 2171	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 ∈ (ℤ≥‘(1 − 1)))
99	15, 98, 22	sylancr 405	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
100	99	reseq2d 4630	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...(𝑁 + 1))) = (𝐺 ↾ ((1...𝑁) ∪ {(𝑁 + 1)})))
101		resundi 4643	. . . . 5 ⊢ (𝐺 ↾ ((1...𝑁) ∪ {(𝑁 + 1)})) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)}))
102	100, 101	syl6req 2130	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → ((𝐺 ↾ (1...𝑁)) ∪ (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 ∈ ℕ0 ∧ (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 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺 = (𝐹 ∪ 𝐻))
106	78, 86, 105	3jca 1118	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻)))
107	71, 106	impbida 560	1 ⊢ (𝑁 ∈ ℕ0 → ((𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 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   ∈ wcel 1433   ∪ cun 2971   ∩ cin 2972   ⊆ 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  ℕcn 8039  ℕ₀cn0 8288  ℤcz 8351  ℤ_≥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