Theorem fseq1p1m1 13330

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 1193	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐹:(1...𝑁)⟶𝐴)
2		nn0p1nn 12272	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
3	2	adantr 481	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝑁 + 1) ∈ ℕ)
4		simpr2 1194	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐵 ∈ 𝐴)
5		fseq1p1m1.1	. . . . . . . . 9 ⊢ 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}
6		fsng 7009	. . . . . . . . 9 ⊢ (((𝑁 + 1) ∈ ℕ ∧ 𝐵 ∈ 𝐴) → (𝐻:{(𝑁 + 1)}⟶{𝐵} ↔ 𝐻 = {⟨(𝑁 + 1), 𝐵⟩}))
7	5, 6	mpbiri 257	. . . . . . . 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 4742	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → {𝐵} ⊆ 𝐴)
10	8, 9	fssd 6618	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐻:{(𝑁 + 1)}⟶𝐴)
11		fzp1disj 13315	. . . . . . 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 6638	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐹 ∪ 𝐻):((1...𝑁) ∪ {(𝑁 + 1)})⟶𝐴)
14		1z 12350	. . . . . . . 8 ⊢ 1 ∈ ℤ
15		simpl 483	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝑁 ∈ ℕ0)
16		nn0uz 12620	. . . . . . . . . 10 ⊢ ℕ0 = (ℤ≥‘0)
17		1m1e0 12045	. . . . . . . . . . 11 ⊢ (1 − 1) = 0
18	17	fveq2i 6777	. . . . . . . . . 10 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0)
19	16, 18	eqtr4i 2769	. . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘(1 − 1))
20	15, 19	eleqtrdi 2849	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝑁 ∈ (ℤ≥‘(1 − 1)))
21		fzsuc2 13314	. . . . . . . 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 2744	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((1...𝑁) ∪ {(𝑁 + 1)}) = (1...(𝑁 + 1)))
24	23	feq2d 6586	. . . . 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 231	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐹 ∪ 𝐻):(1...(𝑁 + 1))⟶𝐴)
26		simpr3 1195	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐺 = (𝐹 ∪ 𝐻))
27	26	feq1d 6585	. . . 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 256	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
29		ovex 7308	. . . . . 6 ⊢ (𝑁 + 1) ∈ V
30	29	snid 4597	. . . . 5 ⊢ (𝑁 + 1) ∈ {(𝑁 + 1)}
31		fvres 6793	. . . . 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 5890	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = ((𝐹 ∪ 𝐻) ↾ {(𝑁 + 1)}))
34		ffn 6600	. . . . . . . . . . 11 ⊢ (𝐹:(1...𝑁)⟶𝐴 → 𝐹 Fn (1...𝑁))
35		fnresdisj 6552	. . . . . . . . . . 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 231	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐹 ↾ {(𝑁 + 1)}) = ∅)
38	37	uneq1d 4096	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)})) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})))
39		resundir 5906	. . . . . . . 8 ⊢ ((𝐹 ∪ 𝐻) ↾ {(𝑁 + 1)}) = ((𝐹 ↾ {(𝑁 + 1)}) ∪ (𝐻 ↾ {(𝑁 + 1)}))
40		uncom 4087	. . . . . . . . 9 ⊢ (∅ ∪ (𝐻 ↾ {(𝑁 + 1)})) = ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅)
41		un0 4324	. . . . . . . . 9 ⊢ ((𝐻 ↾ {(𝑁 + 1)}) ∪ ∅) = (𝐻 ↾ {(𝑁 + 1)})
42	40, 41	eqtr2i 2767	. . . . . . . 8 ⊢ (𝐻 ↾ {(𝑁 + 1)}) = (∅ ∪ (𝐻 ↾ {(𝑁 + 1)}))
43	38, 39, 42	3eqtr4g 2803	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐹 ∪ 𝐻) ↾ {(𝑁 + 1)}) = (𝐻 ↾ {(𝑁 + 1)}))
44		ffn 6600	. . . . . . . 8 ⊢ (𝐻:{(𝑁 + 1)}⟶𝐴 → 𝐻 Fn {(𝑁 + 1)})
45		fnresdm 6551	. . . . . . . 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 2782	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺 ↾ {(𝑁 + 1)}) = 𝐻)
48	47	fveq1d 6776	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = (𝐻‘(𝑁 + 1)))
49	5	fveq1i 6775	. . . . . . 7 ⊢ (𝐻‘(𝑁 + 1)) = ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1))
50		fvsng 7052	. . . . . . 7 ⊢ (((𝑁 + 1) ∈ ℕ ∧ 𝐵 ∈ 𝐴) → ({⟨(𝑁 + 1), 𝐵⟩}‘(𝑁 + 1)) = 𝐵)
51	49, 50	eqtrid 2790	. . . . . 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 2778	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐺 ↾ {(𝑁 + 1)})‘(𝑁 + 1)) = 𝐵)
54	32, 53	eqtr3id 2792	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺‘(𝑁 + 1)) = 𝐵)
55	26	reseq1d 5890	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐺 ↾ (1...𝑁)) = ((𝐹 ∪ 𝐻) ↾ (1...𝑁)))
56		incom 4135	. . . . . . . 8 ⊢ ({(𝑁 + 1)} ∩ (1...𝑁)) = ((1...𝑁) ∩ {(𝑁 + 1)})
57	56, 12	eqtrid 2790	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ({(𝑁 + 1)} ∩ (1...𝑁)) = ∅)
58		ffn 6600	. . . . . . . 8 ⊢ (𝐻:{(𝑁 + 1)}⟶{𝐵} → 𝐻 Fn {(𝑁 + 1)})
59		fnresdisj 6552	. . . . . . . 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 231	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → (𝐻 ↾ (1...𝑁)) = ∅)
62	61	uneq2d 4097	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁))) = ((𝐹 ↾ (1...𝑁)) ∪ ∅))
63		resundir 5906	. . . . 5 ⊢ ((𝐹 ∪ 𝐻) ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ (𝐻 ↾ (1...𝑁)))
64		un0 4324	. . . . . 6 ⊢ ((𝐹 ↾ (1...𝑁)) ∪ ∅) = (𝐹 ↾ (1...𝑁))
65	64	eqcomi 2747	. . . . 5 ⊢ (𝐹 ↾ (1...𝑁)) = ((𝐹 ↾ (1...𝑁)) ∪ ∅)
66	62, 63, 65	3eqtr4g 2803	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → ((𝐹 ∪ 𝐻) ↾ (1...𝑁)) = (𝐹 ↾ (1...𝑁)))
67		fnresdm 6551	. . . . 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 2783	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐹:(1...𝑁)⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
70	28, 54, 69	3jca 1127	. 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 1193	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺:(1...(𝑁 + 1))⟶𝐴)
72		fzssp1 13299	. . . . 5 ⊢ (1...𝑁) ⊆ (1...(𝑁 + 1))
73		fssres 6640	. . . . 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 1195	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹 = (𝐺 ↾ (1...𝑁)))
76	75	feq1d 6585	. . . 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 256	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐹:(1...𝑁)⟶𝐴)
78		simpr2 1194	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) = 𝐵)
79	2	adantr 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ ℕ)
80		nnuz 12621	. . . . . . 7 ⊢ ℕ = (ℤ≥‘1)
81	79, 80	eleqtrdi 2849	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝑁 + 1) ∈ (ℤ≥‘1))
82		eluzfz2 13264	. . . . . 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	ffvelrnd 6962	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → (𝐺‘(𝑁 + 1)) ∈ 𝐴)
85	78, 84	eqeltrrd 2840	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐵 ∈ 𝐴)
86		ffn 6600	. . . . . . . . 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 7030	. . . . . . . 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 4805	. . . . . . . . 9 ⊢ ((𝐺‘(𝑁 + 1)) = 𝐵 → ⟨(𝑁 + 1), (𝐺‘(𝑁 + 1))⟩ = ⟨(𝑁 + 1), 𝐵⟩)
91	90	sneqd 4573	. . . . . . . 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 2778	. . . . . 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 2797	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐻 = (𝐺 ↾ {(𝑁 + 1)}))
95	75, 94	uneq12d 4098	. . . 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 483	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝑁 ∈ ℕ0)
97	96, 19	eleqtrdi 2849	. . . . . . 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 5891	. . . . 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 5905	. . . . 5 ⊢ (𝐺 ↾ ((1...𝑁) ∪ {(𝑁 + 1)})) = ((𝐺 ↾ (1...𝑁)) ∪ (𝐺 ↾ {(𝑁 + 1)}))
101	99, 100	eqtr2di 2795	. . . 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 6551	. . . . 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 2783	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝐺:(1...(𝑁 + 1))⟶𝐴 ∧ (𝐺‘(𝑁 + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...𝑁)))) → 𝐺 = (𝐹 ∪ 𝐻))
105	77, 85, 104	3jca 1127	. 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 798	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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {csn 4561  ⟨cop 4567   ↾ cres 5591   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  0cc0 10871  1c1 10872   + caddc 10874   − cmin 11205  ℕcn 11973  ℕ₀cn0 12233  ℤcz 12319  ℤ_≥cuz 12582  ...cfz 13239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240

This theorem is referenced by:  fseq1m1p1  13331