Theorem fseq1p1m1 13535

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 12457	. . . . . . . . 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 7091	. . . . . . . . 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 4769	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → {𝐵} ⊆ 𝐴)
10	8, 9	fssd 6687	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐻:{(𝑁 + 1)}⟶𝐴)
11		fzp1disj 13520	. . . . . . 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 6706	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐹 ∪ 𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴)
14		1z 12539	. . . . . . . 8 ⊢ 1 ∈ ℤ
15		simpl 482	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝑁 ∈ ℕ0)
16		nn0uz 12811	. . . . . . . . . 10 ⊢ ℕ0 = (ℤ≥‘0)
17		1m1e0 12234	. . . . . . . . . . 11 ⊢ (1 − 1) = 0
18	17	fveq2i 6843	. . . . . . . . . 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 13519	. . . . . . . 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 6654	. . . . 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 6652	. . . 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 7402	. . . . . 6 ⊢ (𝑁 + 1) ∈ V
30	29	snid 4622	. . . . 5 ⊢ (𝑁 + 1) ∈ {(𝑁 + 1)}
31		fvres 6859	. . . . 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 5938	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = ((𝐹 ∪ 𝐻) ↾ {(𝑁 + 1)}))
34		ffn 6670	. . . . . . . . . . 11 ⊢ (𝐹:(1...𝑁)⟶𝐴 → 𝐹 Fn (1...𝑁))
35		fnresdisj 6620	. . . . . . . . . . 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 4126	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)})) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})))
39		resundir 5954	. . . . . . . 8 ⊢ ((𝐹 ∪ 𝐻) ↾ {(𝑁 + 1)}) = ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)}))
40		uncom 4117	. . . . . . . . 9 ⊢ (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})) = ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅)
41		un0 4353	. . . . . . . . 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 6670	. . . . . . . 8 ⊢ (𝐻:{(𝑁 + 1)}⟶𝐴 → 𝐻 Fn {(𝑁 + 1)})
45		fnresdm 6619	. . . . . . . 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 6842	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐻‘(𝑁 + 1)))
49	5	fveq1i 6841	. . . . . . 7 ⊢ (𝐻‘(𝑁 + 1)) = ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1))
50		fvsng 7136	. . . . . . 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 5938	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺 ↾ (1...𝑁)) = ((𝐹 ∪ 𝐻) ↾ (1...𝑁)))
56		incom 4168	. . . . . . . 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 6670	. . . . . . . 8 ⊢ (𝐻:{(𝑁 + 1)}⟶{𝐵} → 𝐻 Fn {(𝑁 + 1)})
59		fnresdisj 6620	. . . . . . . 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 4127	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁))) = ((𝐹 ↾ (1...𝑁)) ∪ ∅))
63		resundir 5954	. . . . 5 ⊢ ((𝐹 ∪ 𝐻) ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁)))
64		un0 4353	. . . . . 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 6619	. . . . 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 13504	. . . . 5 ⊢ (1...𝑁) ⊆ (1...(𝑁 + 1))
73		fssres 6708	. . . . 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 6652	. . . 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 12812	. . . . . . 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 13469	. . . . . 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 7039	. . . 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 6670	. . . . . . . . 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 7112	. . . . . . . 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 4834	. . . . . . . . 9 ⊢ ((𝐺‘(𝑁 + 1)) = 𝐵 → ⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩ = ⟨(𝑁 + 1), 𝐵⟩)
91	90	sneqd 4597	. . . . . . . 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 4128	. . . 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 5939	. . . . 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 5953	. . . . 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 6619	. . . . 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 3909   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  {csn 4585  ⟨cop 4591   ↾ cres 5633   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  0cc0 11044  1c1 11045   + caddc 11047   − cmin 11381  ℕcn 12162  ℕ₀cn0 12418  ℤcz 12505  ℤ_≥cuz 12769  ...cfz 13444

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 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-n0 12419  df-z 12506  df-uz 12770  df-fz 13445

This theorem is referenced by:  fseq1m1p1  13536