2ffzoeq - Mathbox for Alexander van der Vekens

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Theorem 2ffzoeq 43522

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 2825	. . . . . . . . . . . 12 ⊢ (𝐹 = 𝑃 → (𝐹 = ∅ ↔ 𝑃 = ∅))
2	1	anbi1d 631	. . . . . . . . . . 11 ⊢ (𝐹 = 𝑃 → ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
3		f0bi 6556	. . . . . . . . . . . . 13 ⊢ (𝑃:∅⟶𝑌 ↔ 𝑃 = ∅)
4		ffn 6508	. . . . . . . . . . . . . 14 ⊢ (𝑃:(0..^𝑁)⟶𝑌 → 𝑃 Fn (0..^𝑁))
5		ffn 6508	. . . . . . . . . . . . . 14 ⊢ (𝑃:∅⟶𝑌 → 𝑃 Fn ∅)
6		fndmu 6452	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → (0..^𝑁) = ∅)
7		0z 11986	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℤ
8		nn0z 11999	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
9	8	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
10		fzon 13052	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
11	7, 9, 10	sylancr 589	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
12		nn0ge0 11916	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
13		0red 10638	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ ℝ)
14		nn0re 11900	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
15	13, 14	letri3d 10776	. . . . . . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 ≤ 0 → 0 = 𝑁))
18	17	adantl 484	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . 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 409	. . . . . . . . . . 11 ⊢ ((𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 = 𝑁))
25	2, 24	syl6bi 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 7158	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (0..^𝑀) = (0..^0))
29		fzo0 13055	. . . . . . . . . . . 12 ⊢ (0..^0) = ∅
30	28, 29	syl6eq 2872	. . . . . . . . . . 11 ⊢ (𝑀 = 0 → (0..^𝑀) = ∅)
31	30	feq2d 6494	. . . . . . . . . 10 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹:∅⟶𝑋))
32		f0bi 6556	. . . . . . . . . 10 ⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅)
33	31, 32	syl6bb 289	. . . . . . . . 9 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹 = ∅))
34	33	anbi1d 631	. . . . . . . 8 ⊢ (𝑀 = 0 → ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
35		eqeq1 2825	. . . . . . . . . 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 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 410	. . . . 5 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝐹 = 𝑃 → 𝑀 = 𝑁)))
41	40	impcom 410	. . . 4 ⊢ ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 → 𝑀 = 𝑁))
42	28	feq2d 6494	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹:(0..^0)⟶𝑋))
43	29	feq2i 6500	. . . . . . . . . . . . 13 ⊢ (𝐹:(0..^0)⟶𝑋 ↔ 𝐹:∅⟶𝑋)
44	43, 32	bitri 277	. . . . . . . . . . . 12 ⊢ (𝐹:(0..^0)⟶𝑋 ↔ 𝐹 = ∅)
45	42, 44	syl6bb 289	. . . . . . . . . . 11 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹 = ∅))
46	45	adantr 483	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹 = ∅))
47		eqeq1 2825	. . . . . . . . . . . 12 ⊢ (𝑀 = 𝑁 → (𝑀 = 0 ↔ 𝑁 = 0))
48	47	biimpac 481	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → 𝑁 = 0)
49		oveq2 7158	. . . . . . . . . . . . 13 ⊢ (𝑁 = 0 → (0..^𝑁) = (0..^0))
50	49	feq2d 6494	. . . . . . . . . . . 12 ⊢ (𝑁 = 0 → (𝑃:(0..^𝑁)⟶𝑌 ↔ 𝑃:(0..^0)⟶𝑌))
51	29	feq2i 6500	. . . . . . . . . . . . 13 ⊢ (𝑃:(0..^0)⟶𝑌 ↔ 𝑃:∅⟶𝑌)
52	51, 3	bitri 277	. . . . . . . . . . . 12 ⊢ (𝑃:(0..^0)⟶𝑌 ↔ 𝑃 = ∅)
53	50, 52	syl6bb 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 632	. . . . . . . . 9 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝐹 = ∅ ∧ 𝑃 = ∅)))
56		eqtr3 2843	. . . . . . . . 9 ⊢ ((𝐹 = ∅ ∧ 𝑃 = ∅) → 𝐹 = 𝑃)
57	55, 56	syl6bi 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 418	. . . . . 6 ⊢ ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → (𝑀 = 0 → (𝑀 = 𝑁 → 𝐹 = 𝑃)))
60	59	adantl 484	. . . . 5 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝑀 = 𝑁 → 𝐹 = 𝑃)))
61	60	impcom 410	. . . 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 4455	. . . . . 6 ⊢ ∀𝑖 ∈ ∅ (𝐹‘𝑖) = (𝑃‘𝑖)
64	30	raleqdv 3415	. . . . . 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 534	. . . 4 ⊢ (𝑀 = 0 → (𝑀 = 𝑁 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
67	66	adantr 483	. . 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 6508	. . . . . . 7 ⊢ (𝐹:(0..^𝑀)⟶𝑋 → 𝐹 Fn (0..^𝑀))
70	69, 4	anim12i 614	. . . . . 6 ⊢ ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
71	70	adantl 484	. . . . 5 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
72	71	adantl 484	. . . 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 6797	. . . 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 3017	. . . . . 6 ⊢ (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
76		elnnne0 11905	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℕ₀ ∧ 𝑀 ≠ 0))
77		0zd 11987	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 ∈ ℤ)
78		nnz 11998	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
79		nngt0 11662	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 < 𝑀)
80	77, 78, 79	3jca 1124	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
81	80	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
82		fzoopth 43521	. . . . . . . . . . . . 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 487	. . . . . . . . . . . 12 ⊢ ((0 = 0 ∧ 𝑀 = 𝑁) → 𝑀 = 𝑁)
85	83, 84	syl6bi 255	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((0..^𝑀) = (0..^𝑁) → 𝑀 = 𝑁))
86	85	anim1d 612	. . . . . . . . . 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 7158	. . . . . . . . . . 11 ⊢ (𝑀 = 𝑁 → (0..^𝑀) = (0..^𝑁))
88	87	anim1i 616	. . . . . . . . . 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 415	. . . . . . . 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 454	. . . . . 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	syl5bir 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 483	. . . 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 410	. . 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 810	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 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∅c0 4290 class class class wbr 5058 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 0cc0 10531 < clt 10669 ≤ cle 10670 ℕcn 11632 ℕ₀cn0 11891 ℤcz 11975 ..^cfzo 13027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028

This theorem is referenced by: (None)

W3C validator