2ffzoeq - Mathbox for Alexander van der Vekens

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Theorem 2ffzoeq 47358

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 2735	. . . . . . . . . . . 12 ⊢ (𝐹 = 𝑃 → (𝐹 = ∅ ↔ 𝑃 = ∅))
2	1	anbi1d 631	. . . . . . . . . . 11 ⊢ (𝐹 = 𝑃 → ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
3		f0bi 6701	. . . . . . . . . . . . 13 ⊢ (𝑃:∅⟶𝑌 ↔ 𝑃 = ∅)
4		ffn 6646	. . . . . . . . . . . . . 14 ⊢ (𝑃:(0..^𝑁)⟶𝑌 → 𝑃 Fn (0..^𝑁))
5		ffn 6646	. . . . . . . . . . . . . 14 ⊢ (𝑃:∅⟶𝑌 → 𝑃 Fn ∅)
6		fndmu 6583	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → (0..^𝑁) = ∅)
7		0z 12474	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℤ
8		nn0z 12488	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
9	8	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
10		fzon 13575	. . . . . . . . . . . . . . . . . 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 12401	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
13		0red 11110	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ ℝ)
14		nn0re 12385	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
15	13, 14	letri3d 11250	. . . . . . . . . . . . . . . . . . . 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 7349	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (0..^𝑀) = (0..^0))
29		fzo0 13578	. . . . . . . . . . . 12 ⊢ (0..^0) = ∅
30	28, 29	eqtrdi 2782	. . . . . . . . . . 11 ⊢ (𝑀 = 0 → (0..^𝑀) = ∅)
31	30	feq2d 6630	. . . . . . . . . 10 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹:∅⟶𝑋))
32		f0bi 6701	. . . . . . . . . 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 2735	. . . . . . . . . 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 6630	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋 ↔ 𝐹:(0..^0)⟶𝑋))
43	29	feq2i 6638	. . . . . . . . . . . . 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 2735	. . . . . . . . . . . 12 ⊢ (𝑀 = 𝑁 → (𝑀 = 0 ↔ 𝑁 = 0))
48	47	biimpac 478	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝑀 = 𝑁) → 𝑁 = 0)
49		oveq2 7349	. . . . . . . . . . . . 13 ⊢ (𝑁 = 0 → (0..^𝑁) = (0..^0))
50	49	feq2d 6630	. . . . . . . . . . . 12 ⊢ (𝑁 = 0 → (𝑃:(0..^𝑁)⟶𝑌 ↔ 𝑃:(0..^0)⟶𝑌))
51	29	feq2i 6638	. . . . . . . . . . . . 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 2753	. . . . . . . . 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 4458	. . . . . 6 ⊢ ∀𝑖 ∈ ∅ (𝐹‘𝑖) = (𝑃‘𝑖)
64	30	raleqdv 3292	. . . . . 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 6646	. . . . . . 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 6960	. . . 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 2929	. . . . . 6 ⊢ (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
76		elnnne0 12390	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℕ₀ ∧ 𝑀 ≠ 0))
77		0zd 12475	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 ∈ ℤ)
78		nnz 12484	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
79		nngt0 12151	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ → 0 < 𝑀)
80	77, 78, 79	3jca 1128	. . . . . . . . . . . . . 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 13657	. . . . . . . . . . . . 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 7349	. . . . . . . . . . 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 811	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 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∅c0 4278 class class class wbr 5086 Fn wfn 6471 ⟶wf 6472 ‘cfv 6476 (class class class)co 7341 0cc0 11001 < clt 11141 ≤ cle 11142 ℕcn 12120 ℕ₀cn0 12376 ℤcz 12463 ..^cfzo 13549

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-nn 12121 df-n0 12377 df-z 12464 df-uz 12728 df-fz 13403 df-fzo 13550

This theorem is referenced by: (None)

W3C validator