2ffzoeq - Mathbox for Alexander van der Vekens

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Theorem 2ffzoeq 48216

Description: Two functions over a half-open range of nonnegative integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 1-Jul-2018.)

**Assertion**
Ref	Expression
2ffzoeq	⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))

Distinct variable groups: 𝑖,𝐹 𝑖,𝑀 𝑃,𝑖 Allowed substitution hints: 𝑁(𝑖) 𝑋(𝑖) 𝑌(𝑖)

**Proof of Theorem 2ffzoeq**
Step	Hyp	Ref	Expression
1		eqeq1 2764	. . . . . . . . . . . 12 ⊢ (𝐹 = 𝑃 → (𝐹 = ∅ ↔ 𝑃 = ∅))
2	1	anbi1d 643	. . . . . . . . . . 11 ⊢ (𝐹 = 𝑃 → ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
3		f0bi 6758	. . . . . . . . . . . . 13 ⊢ (𝑃:∅⟶𝑌 ↔ 𝑃 = ∅)
4		ffn 6702	. . . . . . . . . . . . . 14 ⊢ (𝑃:(0..^𝑁)⟶𝑌 → 𝑃 Fn (0..^𝑁))
5		ffn 6702	. . . . . . . . . . . . . 14 ⊢ (𝑃:∅⟶𝑌 → 𝑃 Fn ∅)
6		fndmu 6639	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → (0..^𝑁) = ∅)
7		0z 12626	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℤ
8		nn0z 12639	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
9	8	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
10		fzon 13736	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
11	7, 9, 10	sylancr 599	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
12		nn0ge0 12553	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
13		0red 11235	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ ℝ)
14		nn0re 12537	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
15	13, 14	letri3d 11376	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → (0 = 𝑁 ↔ (0 ≤ 𝑁 ∧ 𝑁 ≤ 0)))
16	15	biimprd 251	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → ((0 ≤ 𝑁 ∧ 𝑁 ≤ 0) → 0 = 𝑁))
17	12, 16	mpand 708	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 ≤ 0 → 0 = 𝑁))
18	17	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ 0 → 0 = 𝑁))
19	11, 18	sylbird 263	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((0..^𝑁) = ∅ → 0 = 𝑁))
20	6, 19	syl5com 32	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 = 𝑁))
21	20	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝑃 Fn (0..^𝑁) → (𝑃 Fn ∅ → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 = 𝑁)))
22	4, 5, 21	syl2imc 42	. . . . . . . . . . . . 13 ⊢ (𝑃:∅⟶𝑌 → (𝑃:(0..^𝑁)⟶𝑌 → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 = 𝑁)))
23	3, 22	sylbir 238	. . . . . . . . . . . 12 ⊢ (𝑃 = ∅ → (𝑃:(0..^𝑁)⟶𝑌 → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 = 𝑁)))
24	23	imp 412	. . . . . . . . . . 11 ⊢ ((𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 = 𝑁))
25	2, 24	biimtrdi 256	. . . . . . . . . 10 ⊢ (𝐹 = 𝑃 → ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 = 𝑁)))
26	25	com3l 90	. . . . . . . . 9 ⊢ ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐹 = 𝑃 → 0 = 𝑁)))
27	26	a1i 11	. . . . . . . 8 ⊢ (𝑀 = 0 → ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐹 = 𝑃 → 0 = 𝑁))))
28		oveq2 7421	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (0..^𝑀) = (0..^0))
29		fzo0 13739	. . . . . . . . . . . 12 ⊢ (0..^0) = ∅
30	28, 29	eqtrdi 2811	. . . . . . . . . . 11 ⊢ (𝑀 = 0 → (0..^𝑀) = ∅)
31	30	feq2d 6686	. . . . . . . . . 10 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹:∅⟶𝑋))
32		f0bi 6758	. . . . . . . . . 10 ⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅)
33	31, 32	bitrdi 290	. . . . . . . . 9 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹 = ∅))
34	33	anbi1d 643	. . . . . . . 8 ⊢ (𝑀 = 0 → ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
35		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑀 = 0 → (𝑀 = 𝑁 ↔ 0 = 𝑁))
36	35	imbi2d 343	. . . . . . . . 9 ⊢ (𝑀 = 0 → ((𝐹 = 𝑃 → 𝑀 = 𝑁) ↔ (𝐹 = 𝑃 → 0 = 𝑁)))
37	36	imbi2d 343	. . . . . . . 8 ⊢ (𝑀 = 0 → (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐹 = 𝑃 → 𝑀 = 𝑁)) ↔ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐹 = 𝑃 → 0 = 𝑁))))
38	27, 34, 37	3imtr4d 297	. . . . . . 7 ⊢ (𝑀 = 0 → ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐹 = 𝑃 → 𝑀 = 𝑁))))
39	38	com3l 90	. . . . . 6 ⊢ ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 = 0 → (𝐹 = 𝑃 → 𝑀 = 𝑁))))
40	39	impcom 413	. . . . 5 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝐹 = 𝑃 → 𝑀 = 𝑁)))
41	40	impcom 413	. . . 4 ⊢ ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 → 𝑀 = 𝑁))
42	28	feq2d 6686	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹:(0..^0)⟶𝑋))
43	29	feq2i 6694	. . . . . . . . . . . . 13 ⊢ (𝐹:(0..^0)⟶𝑋 ↔ 𝐹:∅⟶𝑋)
44	43, 32	bitri 278	. . . . . . . . . . . 12 ⊢ (𝐹:(0..^0)⟶𝑋 ↔ 𝐹 = ∅)
45	42, 44	bitrdi 290	. . . . . . . . . . 11 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹 = ∅))
46	45	adantr 486	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹 = ∅))
47		eqeq1 2764	. . . . . . . . . . . 12 ⊢ (𝑀 = 𝑁 → (𝑀 = 0 ↔ 𝑁 = 0))
48	47	biimpac 484	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → 𝑁 = 0)
49		oveq2 7421	. . . . . . . . . . . . 13 ⊢ (𝑁 = 0 → (0..^𝑁) = (0..^0))
50	49	feq2d 6686	. . . . . . . . . . . 12 ⊢ (𝑁 = 0 → (𝑃:(0..^𝑁)⟶𝑌 ↔ 𝑃:(0..^0)⟶𝑌))
51	29	feq2i 6694	. . . . . . . . . . . . 13 ⊢ (𝑃:(0..^0)⟶𝑌 ↔ 𝑃:∅⟶𝑌)
52	51, 3	bitri 278	. . . . . . . . . . . 12 ⊢ (𝑃:(0..^0)⟶𝑌 ↔ 𝑃 = ∅)
53	50, 52	bitrdi 290	. . . . . . . . . . 11 ⊢ (𝑁 = 0 → (𝑃:(0..^𝑁)⟶𝑌 ↔ 𝑃 = ∅))
54	48, 53	syl 18	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → (𝑃:(0..^𝑁)⟶𝑌 ↔ 𝑃 = ∅))
55	46, 54	anbi12d 644	. . . . . . . . 9 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝐹 = ∅ ∧ 𝑃 = ∅)))
56		eqtr3 2782	. . . . . . . . 9 ⊢ ((𝐹 = ∅ ∧ 𝑃 = ∅) → 𝐹 = 𝑃)
57	55, 56	biimtrdi 256	. . . . . . . 8 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → 𝐹 = 𝑃))
58	57	com12 33	. . . . . . 7 ⊢ ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 = 0 ∧ 𝑀 = 𝑁) → 𝐹 = 𝑃))
59	58	expd 421	. . . . . 6 ⊢ ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → (𝑀 = 0 → (𝑀 = 𝑁 → 𝐹 = 𝑃)))
60	59	adantl 487	. . . . 5 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝑀 = 𝑁 → 𝐹 = 𝑃)))
61	60	impcom 413	. . . 4 ⊢ ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (𝑀 = 𝑁 → 𝐹 = 𝑃))
62	41, 61	impbid 215	. . 3 ⊢ ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 ↔ 𝑀 = 𝑁))
63		ral0 4454	. . . . . 6 ⊢ ∀𝑖 ∈ ∅ (𝐹‘𝑖) = (𝑃‘𝑖)
64	30	raleqdv 3319	. . . . . 6 ⊢ (𝑀 = 0 → (∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖) ↔ ∀𝑖 ∈ ∅ (𝐹‘𝑖) = (𝑃‘𝑖)))
65	63, 64	mpbiri 261	. . . . 5 ⊢ (𝑀 = 0 → ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))
66	65	biantrud 541	. . . 4 ⊢ (𝑀 = 0 → (𝑀 = 𝑁 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
67	66	adantr 486	. . 3 ⊢ ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (𝑀 = 𝑁 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
68	62, 67	bitrd 282	. 2 ⊢ ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
69		ffn 6702	. . . . . . 7 ⊢ (𝐹:(0..^𝑀)⟶𝑋 → 𝐹 Fn (0..^𝑀))
70	69, 4	anim12i 625	. . . . . 6 ⊢ ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
71	70	adantl 487	. . . . 5 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
72	71	adantl 487	. . . 4 ⊢ ((¬ 𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
73		eqfnfv2 7023	. . . 4 ⊢ ((𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)) → (𝐹 = 𝑃 ↔ ((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
74	72, 73	syl 18	. . 3 ⊢ ((¬ 𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 ↔ ((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
75		df-ne 2956	. . . . . 6 ⊢ (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
76		elnnne0 12542	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℕ₀ ∧ 𝑀 ≠ 0))
77		0zd 12627	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 ∈ ℤ)
78		nnz 12636	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
79		nngt0 12291	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 < 𝑀)
80	77, 78, 79	3jca 1146	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
81	80	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
82		fzoopth 13818	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀) → ((0..^𝑀) = (0..^𝑁) ↔ (0 = 0 ∧ 𝑀 = 𝑁)))
83	81, 82	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((0..^𝑀) = (0..^𝑁) ↔ (0 = 0 ∧ 𝑀 = 𝑁)))
84		simpr 490	. . . . . . . . . . . 12 ⊢ ((0 = 0 ∧ 𝑀 = 𝑁) → 𝑀 = 𝑁)
85	83, 84	biimtrdi 256	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((0..^𝑀) = (0..^𝑁) → 𝑀 = 𝑁))
86	85	anim1d 623	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)) → (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
87		oveq2 7421	. . . . . . . . . . 11 ⊢ (𝑀 = 𝑁 → (0..^𝑀) = (0..^𝑁))
88	87	anim1i 627	. . . . . . . . . 10 ⊢ ((𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)) → ((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)))
89	86, 88	impbid1 228	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
90	89	ex 418	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑁 ∈ ℕ₀ → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)))))
91	76, 90	sylbir 238	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑀 ≠ 0) → (𝑁 ∈ ℕ₀ → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)))))
92	91	impancom 457	. . . . . 6 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 ≠ 0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)))))
93	75, 92	biimtrrid 246	. . . . 5 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (¬ 𝑀 = 0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)))))
94	93	adantr 486	. . . 4 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (¬ 𝑀 = 0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)))))
95	94	impcom 413	. . 3 ⊢ ((¬ 𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
96	74, 95	bitrd 282	. 2 ⊢ ((¬ 𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
97	68, 96	pm2.61ian 824	1 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∅c0 4279 class class class wbr 5103 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 0cc0 11124 < clt 11267 ≤ cle 11268 ℕcn 12257 ℕ₀cn0 12528 ℤcz 12615 ..^cfzo 13709

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-n0 12529 df-z 12616 df-uz 12888 df-fz 13562 df-fzo 13710

This theorem is used by: (None)

W3C validator