Theorem fseq1p1m1 13559

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 1195	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐹:(1...𝑁)⟶𝐴)
2		nn0p1nn 12481	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
3	2	adantr 480	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝑁 + 1) ∈ ℕ)
4		simpr2 1196	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐵 ∈ 𝐴)
5		fseq1p1m1.1	. . . . . . . . 9 ⊢ 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}
6		fsng 7109	. . . . . . . . 9 ⊢ (((𝑁 + 1) ∈ ℕ ∧ 𝐵 ∈ 𝐴) → (𝐻:{(𝑁 + 1)}⟶{𝐵} ↔ 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}))
7	5, 6	mpbiri 258	. . . . . . . 8 ⊢ (((𝑁 + 1) ∈ ℕ ∧ 𝐵 ∈ 𝐴) → 𝐻:{(𝑁 + 1)}⟶{𝐵})
8	3, 4, 7	syl2anc 584	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐻:{(𝑁 + 1)}⟶{𝐵})
9	4	snssd 4773	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → {𝐵} ⊆ 𝐴)
10	8, 9	fssd 6705	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐻:{(𝑁 + 1)}⟶𝐴)
11		fzp1disj 13544	. . . . . . 7 ⊢ ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅
12	11	a1i 11	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((1...𝑁) ∩ {(𝑁 + 1)}) = ∅)
13	1, 10, 12	fun2d 6724	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐹 ∪ 𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴)
14		1z 12563	. . . . . . . 8 ⊢ 1 ∈ ℤ
15		simpl 482	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝑁 ∈ ℕ0)
16		nn0uz 12835	. . . . . . . . . 10 ⊢ ℕ0 = (ℤ≥‘0)
17		1m1e0 12258	. . . . . . . . . . 11 ⊢ (1 − 1) = 0
18	17	fveq2i 6861	. . . . . . . . . 10 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0)
19	16, 18	eqtr4i 2755	. . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘(1 − 1))
20	15, 19	eleqtrdi 2838	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝑁 ∈ (ℤ≥‘(1 − 1)))
21		fzsuc2 13543	. . . . . . . 8 ⊢ ((1 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(1 − 1))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
22	14, 20, 21	sylancr 587	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
23	22	eqcomd 2735	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((1...𝑁) ∪ {(𝑁 + 1)}) = (1...(𝑁 + 1)))
24	23	feq2d 6672	. . . . 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))
25	13, 24	mpbid 232	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐹 ∪ 𝐻):(1...(𝑁 + 1))⟶𝐴)
26		simpr3 1197	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐺 = (𝐹 ∪ 𝐻))
27	26	feq1d 6670	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺:(1...(𝑁 + 1))⟶𝐴 ↔ (𝐹 ∪ 𝐻):(1...(𝑁 + 1))⟶𝐴))
28	25, 27	mpbird 257	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
29		ovex 7420	. . . . . 6 ⊢ (𝑁 + 1) ∈ V
30	29	snid 4626	. . . . 5 ⊢ (𝑁 + 1) ∈ {(𝑁 + 1)}
31		fvres 6877	. . . . 5 ⊢ ((𝑁 + 1) ∈ {(𝑁 + 1)} → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐺‘(𝑁 + 1)))
32	30, 31	ax-mp 5	. . . 4 ⊢ ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐺‘(𝑁 + 1))
33	26	reseq1d 5949	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = ((𝐹 ∪ 𝐻) ↾ {(𝑁 + 1)}))
34		ffn 6688	. . . . . . . . . . 11 ⊢ (𝐹:(1...𝑁)⟶𝐴 → 𝐹 Fn (1...𝑁))
35		fnresdisj 6638	. . . . . . . . . . 11 ⊢ (𝐹 Fn (1...𝑁) → (((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ ↔ (𝐹 ↾ {(𝑁 + 1)}) = ∅))
36	1, 34, 35	3syl 18	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (((1...𝑁) ∩ {(𝑁 + 1)}) = ∅ ↔ (𝐹 ↾ {(𝑁 + 1)}) = ∅))
37	12, 36	mpbid 232	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐹 ↾ {(𝑁 + 1)}) = ∅)
38	37	uneq1d 4130	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)})) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})))
39		resundir 5965	. . . . . . . 8 ⊢ ((𝐹 ∪ 𝐻) ↾ {(𝑁 + 1)}) = ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)}))
40		uncom 4121	. . . . . . . . 9 ⊢ (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})) = ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅)
41		un0 4357	. . . . . . . . 9 ⊢ ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅) = (𝐻 ↾ {(𝑁 + 1)})
42	40, 41	eqtr2i 2753	. . . . . . . 8 ⊢ (𝐻 ↾ {(𝑁 + 1)}) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)}))
43	38, 39, 42	3eqtr4g 2789	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐹 ∪ 𝐻) ↾ {(𝑁 + 1)}) = (𝐻 ↾ {(𝑁 + 1)}))
44		ffn 6688	. . . . . . . 8 ⊢ (𝐻:{(𝑁 + 1)}⟶𝐴 → 𝐻 Fn {(𝑁 + 1)})
45		fnresdm 6637	. . . . . . . 8 ⊢ (𝐻 Fn {(𝑁 + 1)} → (𝐻 ↾ {(𝑁 + 1)}) = 𝐻)
46	10, 44, 45	3syl 18	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐻 ↾ {(𝑁 + 1)}) = 𝐻)
47	33, 43, 46	3eqtrd 2768	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = 𝐻)
48	47	fveq1d 6860	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐻‘(𝑁 + 1)))
49	5	fveq1i 6859	. . . . . . 7 ⊢ (𝐻‘(𝑁 + 1)) = ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1))
50		fvsng 7154	. . . . . . 7 ⊢ (((𝑁 + 1) ∈ ℕ ∧ 𝐵 ∈ 𝐴) → ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1)) = 𝐵)
51	49, 50	eqtrid 2776	. . . . . 6 ⊢ (((𝑁 + 1) ∈ ℕ ∧ 𝐵 ∈ 𝐴) → (𝐻‘(𝑁 + 1)) = 𝐵)
52	3, 4, 51	syl2anc 584	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐻‘(𝑁 + 1)) = 𝐵)
53	48, 52	eqtrd 2764	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = 𝐵)
54	32, 53	eqtr3id 2778	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺‘(𝑁 + 1)) = 𝐵)
55	26	reseq1d 5949	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺 ↾ (1...𝑁)) = ((𝐹 ∪ 𝐻) ↾ (1...𝑁)))
56		incom 4172	. . . . . . . 8 ⊢ ({(𝑁 + 1)} ∩ (1...𝑁)) = ((1...𝑁) ∩ {(𝑁 + 1)})
57	56, 12	eqtrid 2776	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ({(𝑁 + 1)} ∩ (1...𝑁)) = ∅)
58		ffn 6688	. . . . . . . 8 ⊢ (𝐻:{(𝑁 + 1)}⟶{𝐵} → 𝐻 Fn {(𝑁 + 1)})
59		fnresdisj 6638	. . . . . . . 8 ⊢ (𝐻 Fn {(𝑁 + 1)} → (({(𝑁 + 1)} ∩ (1...𝑁)) = ∅ ↔ (𝐻 ↾ (1...𝑁)) = ∅))
60	8, 58, 59	3syl 18	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (({(𝑁 + 1)} ∩ (1...𝑁)) = ∅ ↔ (𝐻 ↾ (1...𝑁)) = ∅))
61	57, 60	mpbid 232	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐻 ↾ (1...𝑁)) = ∅)
62	61	uneq2d 4131	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁))) = ((𝐹 ↾ (1...𝑁)) ∪ ∅))
63		resundir 5965	. . . . 5 ⊢ ((𝐹 ∪ 𝐻) ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁)))
64		un0 4357	. . . . . 6 ⊢ ((𝐹 ↾ (1...𝑁)) ∪ ∅) = (𝐹 ↾ (1...𝑁))
65	64	eqcomi 2738	. . . . 5 ⊢ (𝐹 ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ ∅)
66	62, 63, 65	3eqtr4g 2789	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐹 ∪ 𝐻) ↾ (1...𝑁)) = (𝐹 ↾ (1...𝑁)))
67		fnresdm 6637	. . . . 5 ⊢ (𝐹 Fn (1...𝑁) → (𝐹 ↾ (1...𝑁)) = 𝐹)
68	1, 34, 67	3syl 18	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐹 ↾ (1...𝑁)) = 𝐹)
69	55, 66, 68	3eqtrrd 2769	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
70	28, 54, 69	3jca 1128	. 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))))
71		simpr1 1195	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
72		fzssp1 13528	. . . . 5 ⊢ (1...𝑁) ⊆ (1...(𝑁 + 1))
73		fssres 6726	. . . . 5 ⊢ ((𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (1...𝑁) ⊆ (1...(𝑁 + 1))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶𝐴)
74	71, 72, 73	sylancl 586	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...𝑁)):(1...𝑁)⟶𝐴)
75		simpr3 1197	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
76	75	feq1d 6670	. . . 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))
77	74, 76	mpbird 257	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹:(1...𝑁)⟶𝐴)
78		simpr2 1196	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) = 𝐵)
79	2	adantr 480	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ ℕ)
80		nnuz 12836	. . . . . . 7 ⊢ ℕ = (ℤ≥‘1)
81	79, 80	eleqtrdi 2838	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ (ℤ≥‘1))
82		eluzfz2 13493	. . . . . 6 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘1) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
83	81, 82	syl 17	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ (1...(𝑁 + 1)))
84	71, 83	ffvelcdmd 7057	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) ∈ 𝐴)
85	78, 84	eqeltrrd 2829	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐵 ∈ 𝐴)
86		ffn 6688	. . . . . . . . 9 ⊢ (𝐺:(1...(𝑁 + 1))⟶𝐴 → 𝐺 Fn (1...(𝑁 + 1)))
87	71, 86	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺 Fn (1...(𝑁 + 1)))
88		fnressn 7130	. . . . . . . 8 ⊢ ((𝐺 Fn (1...(𝑁 + 1)) ∧ (𝑁 + 1) ∈ (1...(𝑁 + 1))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩})
89	87, 83, 88	syl2anc 584	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩})
90		opeq2 4838	. . . . . . . . 9 ⊢ ((𝐺‘(𝑁 + 1)) = 𝐵 → ⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩ = ⟨(𝑁 + 1), 𝐵⟩)
91	90	sneqd 4601	. . . . . . . 8 ⊢ ((𝐺‘(𝑁 + 1)) = 𝐵 → {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩} = {⟨(𝑁 + 1), 𝐵⟩})
92	78, 91	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → {⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩} = {⟨(𝑁 + 1), 𝐵⟩})
93	89, 92	eqtrd 2764	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ {(𝑁 + 1)}) = {⟨(𝑁 + 1), 𝐵⟩})
94	5, 93	eqtr4id 2783	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐻 = (𝐺 ↾ {(𝑁 + 1)}))
95	75, 94	uneq12d 4132	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐹 ∪ 𝐻) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)})))
96		simpl 482	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 ∈ ℕ0)
97	96, 19	eleqtrdi 2838	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 ∈ (ℤ≥‘(1 − 1)))
98	14, 97, 21	sylancr 587	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (1...(𝑁 + 1)) = ((1...𝑁) ∪ {(𝑁 + 1)}))
99	98	reseq2d 5950	. . . . 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)})))
100		resundi 5964	. . . . 5 ⊢ (𝐺 ↾ ((1...𝑁) ∪ {(𝑁 + 1)})) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)}))
101	99, 100	eqtr2di 2781	. . . 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))))
102		fnresdm 6637	. . . . 5 ⊢ (𝐺 Fn (1...(𝑁 + 1)) → (𝐺 ↾ (1...(𝑁 + 1))) = 𝐺)
103	71, 86, 102	3syl 18	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺 ↾ (1...(𝑁 + 1))) = 𝐺)
104	95, 101, 103	3eqtrrd 2769	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺 = (𝐹 ∪ 𝐻))
105	77, 85, 104	3jca 1128	. 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)))
106	70, 105	impbida 800	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   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∪ cun 3912   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  {csn 4589  ⟨cop 4595   ↾ cres 5640   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  0cc0 11068  1c1 11069   + caddc 11071   − cmin 11405  ℕcn 12186  ℕ₀cn0 12442  ℤcz 12529  ℤ_≥cuz 12793  ...cfz 13468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469

This theorem is referenced by:  fseq1m1p1  13560