2ffzoeq - Mathbox for Alexander van der Vekens

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Theorem 2ffzoeq 47339

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 2741	. . . . . . . . . . . 12 ⊢ (𝐹 = 𝑃 → (𝐹 = ∅ ↔ 𝑃 = ∅))
2	1	anbi1d 631	. . . . . . . . . . 11 ⊢ (𝐹 = 𝑃 → ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
3		f0bi 6791	. . . . . . . . . . . . 13 ⊢ (𝑃:∅⟶𝑌 ↔ 𝑃 = ∅)
4		ffn 6736	. . . . . . . . . . . . . 14 ⊢ (𝑃:(0..^𝑁)⟶𝑌 → 𝑃 Fn (0..^𝑁))
5		ffn 6736	. . . . . . . . . . . . . 14 ⊢ (𝑃:∅⟶𝑌 → 𝑃 Fn ∅)
6		fndmu 6675	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → (0..^𝑁) = ∅)
7		0z 12624	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℤ
8		nn0z 12638	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
9	8	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
10		fzon 13720	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
11	7, 9, 10	sylancr 587	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
12		nn0ge0 12551	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
13		0red 11264	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ ℝ)
14		nn0re 12535	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
15	13, 14	letri3d 11403	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → (0 = 𝑁 ↔ (0 ≤ 𝑁 ∧ 𝑁 ≤ 0)))
16	15	biimprd 248	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → ((0 ≤ 𝑁 ∧ 𝑁 ≤ 0) → 0 = 𝑁))
17	12, 16	mpand 695	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 ≤ 0 → 0 = 𝑁))
18	17	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ 0 → 0 = 𝑁))
19	11, 18	sylbird 260	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . 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 235	. . . . . . . . . . . 12 ⊢ (𝑃 = ∅ → (𝑃:(0..^𝑁)⟶𝑌 → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 = 𝑁)))
24	23	imp 406	. . . . . . . . . . 11 ⊢ ((𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 = 𝑁))
25	2, 24	biimtrdi 253	. . . . . . . . . 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 7439	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (0..^𝑀) = (0..^0))
29		fzo0 13723	. . . . . . . . . . . 12 ⊢ (0..^0) = ∅
30	28, 29	eqtrdi 2793	. . . . . . . . . . 11 ⊢ (𝑀 = 0 → (0..^𝑀) = ∅)
31	30	feq2d 6722	. . . . . . . . . 10 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹:∅⟶𝑋))
32		f0bi 6791	. . . . . . . . . 10 ⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅)
33	31, 32	bitrdi 287	. . . . . . . . 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 2741	. . . . . . . . . 10 ⊢ (𝑀 = 0 → (𝑀 = 𝑁 ↔ 0 = 𝑁))
36	35	imbi2d 340	. . . . . . . . 9 ⊢ (𝑀 = 0 → ((𝐹 = 𝑃 → 𝑀 = 𝑁) ↔ (𝐹 = 𝑃 → 0 = 𝑁)))
37	36	imbi2d 340	. . . . . . . 8 ⊢ (𝑀 = 0 → (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐹 = 𝑃 → 𝑀 = 𝑁)) ↔ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝐹 = 𝑃 → 0 = 𝑁))))
38	27, 34, 37	3imtr4d 294	. . . . . . 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 407	. . . . 5 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝐹 = 𝑃 → 𝑀 = 𝑁)))
41	40	impcom 407	. . . 4 ⊢ ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 → 𝑀 = 𝑁))
42	28	feq2d 6722	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹:(0..^0)⟶𝑋))
43	29	feq2i 6728	. . . . . . . . . . . . 13 ⊢ (𝐹:(0..^0)⟶𝑋 ↔ 𝐹:∅⟶𝑋)
44	43, 32	bitri 275	. . . . . . . . . . . 12 ⊢ (𝐹:(0..^0)⟶𝑋 ↔ 𝐹 = ∅)
45	42, 44	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹 = ∅))
46	45	adantr 480	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹 = ∅))
47		eqeq1 2741	. . . . . . . . . . . 12 ⊢ (𝑀 = 𝑁 → (𝑀 = 0 ↔ 𝑁 = 0))
48	47	biimpac 478	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → 𝑁 = 0)
49		oveq2 7439	. . . . . . . . . . . . 13 ⊢ (𝑁 = 0 → (0..^𝑁) = (0..^0))
50	49	feq2d 6722	. . . . . . . . . . . 12 ⊢ (𝑁 = 0 → (𝑃:(0..^𝑁)⟶𝑌 ↔ 𝑃:(0..^0)⟶𝑌))
51	29	feq2i 6728	. . . . . . . . . . . . 13 ⊢ (𝑃:(0..^0)⟶𝑌 ↔ 𝑃:∅⟶𝑌)
52	51, 3	bitri 275	. . . . . . . . . . . 12 ⊢ (𝑃:(0..^0)⟶𝑌 ↔ 𝑃 = ∅)
53	50, 52	bitrdi 287	. . . . . . . . . . 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 2763	. . . . . . . . 9 ⊢ ((𝐹 = ∅ ∧ 𝑃 = ∅) → 𝐹 = 𝑃)
57	55, 56	biimtrdi 253	. . . . . . . 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 415	. . . . . 6 ⊢ ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → (𝑀 = 0 → (𝑀 = 𝑁 → 𝐹 = 𝑃)))
60	59	adantl 481	. . . . 5 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝑀 = 𝑁 → 𝐹 = 𝑃)))
61	60	impcom 407	. . . 4 ⊢ ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (𝑀 = 𝑁 → 𝐹 = 𝑃))
62	41, 61	impbid 212	. . 3 ⊢ ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 ↔ 𝑀 = 𝑁))
63		ral0 4513	. . . . . 6 ⊢ ∀𝑖 ∈ ∅ (𝐹‘𝑖) = (𝑃‘𝑖)
64	30	raleqdv 3326	. . . . . 6 ⊢ (𝑀 = 0 → (∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖) ↔ ∀𝑖 ∈ ∅ (𝐹‘𝑖) = (𝑃‘𝑖)))
65	63, 64	mpbiri 258	. . . . 5 ⊢ (𝑀 = 0 → ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))
66	65	biantrud 531	. . . 4 ⊢ (𝑀 = 0 → (𝑀 = 𝑁 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹‘𝑖) = (𝑃‘𝑖))))
67	66	adantr 480	. . 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 279	. 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 6736	. . . . . . 7 ⊢ (𝐹:(0..^𝑀)⟶𝑋 → 𝐹 Fn (0..^𝑀))
70	69, 4	anim12i 613	. . . . . 6 ⊢ ((𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
71	70	adantl 481	. . . . 5 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝐹:(0..^𝑀)⟶𝑋 ∧ 𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
72	71	adantl 481	. . . 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 7052	. . . 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 2941	. . . . . 6 ⊢ (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
76		elnnne0 12540	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℕ₀ ∧ 𝑀 ≠ 0))
77		0zd 12625	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 ∈ ℤ)
78		nnz 12634	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
79		nngt0 12297	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 < 𝑀)
80	77, 78, 79	3jca 1129	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
81	80	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
82		fzoopth 13801	. . . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((0 = 0 ∧ 𝑀 = 𝑁) → 𝑀 = 𝑁)
85	83, 84	biimtrdi 253	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → ((0..^𝑀) = (0..^𝑁) → 𝑀 = 𝑁))
86	85	anim1d 611	. . . . . . . . . 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 7439	. . . . . . . . . . 11 ⊢ (𝑀 = 𝑁 → (0..^𝑀) = (0..^𝑁))
88	87	anim1i 615	. . . . . . . . . 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 225	. . . . . . . . 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 412	. . . . . . . 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 235	. . . . . . 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 451	. . . . . 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 243	. . . . 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 480	. . . 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 407	. . 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 279	. 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 812	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∅c0 4333 class class class wbr 5143 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 < clt 11295 ≤ cle 11296 ℕcn 12266 ℕ₀cn0 12526 ℤcz 12613 ..^cfzo 13694

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695

This theorem is referenced by: (None)

W3C validator