2ffzoeq - Mathbox for Alexander van der Vekens

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Theorem 2ffzoeq 47883

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 2765	. . . . . . . . . . . 12 ⊢ (𝐹 = 𝑃 → (𝐹 = ∅ ↔ 𝑃 = ∅))
2	1	anbi1d 640	. . . . . . . . . . 11 ⊢ (𝐹 = 𝑃 → ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
3		f0bi 6742	. . . . . . . . . . . . 13 ⊢ (𝑃:∅⟶𝑌 ↔ 𝑃 = ∅)
4		ffn 6686	. . . . . . . . . . . . . 14 ⊢ (𝑃:(0..^𝑁)⟶𝑌 → 𝑃 Fn (0..^𝑁))
5		ffn 6686	. . . . . . . . . . . . . 14 ⊢ (𝑃:∅⟶𝑌 → 𝑃 Fn ∅)
6		fndmu 6623	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → (0..^𝑁) = ∅)
7		0z 12573	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℤ
8		nn0z 12586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
9	8	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
10		fzon 13680	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
11	7, 9, 10	sylancr 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
12		nn0ge0 12500	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
13		0red 11178	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ ℝ)
14		nn0re 12484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
15	13, 14	letri3d 11319	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → (0 = 𝑁 ↔ (0 ≤ 𝑁 ∧ 𝑁 ≤ 0)))
16	15	biimprd 250	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → ((0 ≤ 𝑁 ∧ 𝑁 ≤ 0) → 0 = 𝑁))
17	12, 16	mpand 705	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 ≤ 0 → 0 = 𝑁))
18	17	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ 0 → 0 = 𝑁))
19	11, 18	sylbird 262	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((0..^𝑁) = ∅ → 0 = 𝑁))
20	6, 19	syl5com 31	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 = 𝑁))
21	20	ex 416	. . . . . . . . . . . . . 14 ⊢ (𝑃 Fn (0..^𝑁) → (𝑃 Fn ∅ → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 = 𝑁)))
22	4, 5, 21	syl2imc 41	. . . . . . . . . . . . 13 ⊢ (𝑃:∅⟶𝑌 → (𝑃:(0..^𝑁)⟶𝑌 → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 = 𝑁)))
23	3, 22	sylbir 237	. . . . . . . . . . . 12 ⊢ (𝑃 = ∅ → (𝑃:(0..^𝑁)⟶𝑌 → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 = 𝑁)))
24	23	imp 410	. . . . . . . . . . 11 ⊢ ((𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 = 𝑁))
25	2, 24	biimtrdi 255	. . . . . . . . . 10 ⊢ (𝐹 = 𝑃 → ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 = 𝑁)))
26	25	com3l 89	. . . . . . . . 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 7399	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (0..^𝑀) = (0..^0))
29		fzo0 13683	. . . . . . . . . . . 12 ⊢ (0..^0) = ∅
30	28, 29	eqtrdi 2812	. . . . . . . . . . 11 ⊢ (𝑀 = 0 → (0..^𝑀) = ∅)
31	30	feq2d 6670	. . . . . . . . . 10 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹:∅⟶𝑋))
32		f0bi 6742	. . . . . . . . . 10 ⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅)
33	31, 32	bitrdi 289	. . . . . . . . 9 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹 = ∅))
34	33	anbi1d 640	. . . . . . . 8 ⊢ (𝑀 = 0 → ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
35		eqeq1 2765	. . . . . . . . . 10 ⊢ (𝑀 = 0 → (𝑀 = 𝑁 ↔ 0 = 𝑁))
36	35	imbi2d 342	. . . . . . . . 9 ⊢ (𝑀 = 0 → ((𝐹 = 𝑃 → 𝑀 = 𝑁) ↔ (𝐹 = 𝑃 → 0 = 𝑁)))
37	36	imbi2d 342	. . . . . . . 8 ⊢ (𝑀 = 0 → (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐹 = 𝑃 → 𝑀 = 𝑁)) ↔ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐹 = 𝑃 → 0 = 𝑁))))
38	27, 34, 37	3imtr4d 296	. . . . . . 7 ⊢ (𝑀 = 0 → ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐹 = 𝑃 → 𝑀 = 𝑁))))
39	38	com3l 89	. . . . . 6 ⊢ ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 = 0 → (𝐹 = 𝑃 → 𝑀 = 𝑁))))
40	39	impcom 411	. . . . 5 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝐹 = 𝑃 → 𝑀 = 𝑁)))
41	40	impcom 411	. . . 4 ⊢ ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 → 𝑀 = 𝑁))
42	28	feq2d 6670	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹:(0..^0)⟶𝑋))
43	29	feq2i 6678	. . . . . . . . . . . . 13 ⊢ (𝐹:(0..^0)⟶𝑋 ↔ 𝐹:∅⟶𝑋)
44	43, 32	bitri 277	. . . . . . . . . . . 12 ⊢ (𝐹:(0..^0)⟶𝑋 ↔ 𝐹 = ∅)
45	42, 44	bitrdi 289	. . . . . . . . . . 11 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹 = ∅))
46	45	adantr 484	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹 = ∅))
47		eqeq1 2765	. . . . . . . . . . . 12 ⊢ (𝑀 = 𝑁 → (𝑀 = 0 ↔ 𝑁 = 0))
48	47	biimpac 482	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → 𝑁 = 0)
49		oveq2 7399	. . . . . . . . . . . . 13 ⊢ (𝑁 = 0 → (0..^𝑁) = (0..^0))
50	49	feq2d 6670	. . . . . . . . . . . 12 ⊢ (𝑁 = 0 → (𝑃:(0..^𝑁)⟶𝑌 ↔ 𝑃:(0..^0)⟶𝑌))
51	29	feq2i 6678	. . . . . . . . . . . . 13 ⊢ (𝑃:(0..^0)⟶𝑌 ↔ 𝑃:∅⟶𝑌)
52	51, 3	bitri 277	. . . . . . . . . . . 12 ⊢ (𝑃:(0..^0)⟶𝑌 ↔ 𝑃 = ∅)
53	50, 52	bitrdi 289	. . . . . . . . . . 11 ⊢ (𝑁 = 0 → (𝑃:(0..^𝑁)⟶𝑌 ↔ 𝑃 = ∅))
54	48, 53	syl 17	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → (𝑃:(0..^𝑁)⟶𝑌 ↔ 𝑃 = ∅))
55	46, 54	anbi12d 641	. . . . . . . . 9 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝐹 = ∅ ∧ 𝑃 = ∅)))
56		eqtr3 2783	. . . . . . . . 9 ⊢ ((𝐹 = ∅ ∧ 𝑃 = ∅) → 𝐹 = 𝑃)
57	55, 56	biimtrdi 255	. . . . . . . 8 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → 𝐹 = 𝑃))
58	57	com12 32	. . . . . . 7 ⊢ ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 = 0 ∧ 𝑀 = 𝑁) → 𝐹 = 𝑃))
59	58	expd 419	. . . . . 6 ⊢ ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → (𝑀 = 0 → (𝑀 = 𝑁 → 𝐹 = 𝑃)))
60	59	adantl 485	. . . . 5 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝑀 = 𝑁 → 𝐹 = 𝑃)))
61	60	impcom 411	. . . 4 ⊢ ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (𝑀 = 𝑁 → 𝐹 = 𝑃))
62	41, 61	impbid 214	. . 3 ⊢ ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 ↔ 𝑀 = 𝑁))
63		ral0 4449	. . . . . 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 260	. . . . 5 ⊢ (𝑀 = 0 → ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))
66	65	biantrud 539	. . . 4 ⊢ (𝑀 = 0 → (𝑀 = 𝑁 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
67	66	adantr 484	. . 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 281	. 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 6686	. . . . . . 7 ⊢ (𝐹:(0..^𝑀)⟶𝑋 → 𝐹 Fn (0..^𝑀))
70	69, 4	anim12i 622	. . . . . 6 ⊢ ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
71	70	adantl 485	. . . . 5 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
72	71	adantl 485	. . . 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 7007	. . . 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 17	. . 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 2957	. . . . . 6 ⊢ (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
76		elnnne0 12489	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℕ₀ ∧ 𝑀 ≠ 0))
77		0zd 12574	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 ∈ ℤ)
78		nnz 12583	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
79		nngt0 12238	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 < 𝑀)
80	77, 78, 79	3jca 1140	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
81	80	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
82		fzoopth 13762	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀) → ((0..^𝑀) = (0..^𝑁) ↔ (0 = 0 ∧ 𝑀 = 𝑁)))
83	81, 82	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((0..^𝑀) = (0..^𝑁) ↔ (0 = 0 ∧ 𝑀 = 𝑁)))
84		simpr 488	. . . . . . . . . . . 12 ⊢ ((0 = 0 ∧ 𝑀 = 𝑁) → 𝑀 = 𝑁)
85	83, 84	biimtrdi 255	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((0..^𝑀) = (0..^𝑁) → 𝑀 = 𝑁))
86	85	anim1d 620	. . . . . . . . . 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 7399	. . . . . . . . . . 11 ⊢ (𝑀 = 𝑁 → (0..^𝑀) = (0..^𝑁))
88	87	anim1i 624	. . . . . . . . . 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 227	. . . . . . . . 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 416	. . . . . . . 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 237	. . . . . . 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 455	. . . . . 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 245	. . . . 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 484	. . . 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 411	. . 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 281	. 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 821	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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 ∅c0 4283 class class class wbr 5097 Fn wfn 6511 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 0cc0 11067 < clt 11210 ≤ cle 11211 ℕcn 12204 ℕ₀cn0 12475 ℤcz 12562 ..^cfzo 13653

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-er 8672 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-n0 12476 df-z 12563 df-uz 12834 df-fz 13507 df-fzo 13654

This theorem is referenced by: (None)

W3C validator