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Theorem 2ffzoeq 43676
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 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
Distinct variable groups:   𝑖,𝐹   𝑖,𝑀   𝑃,𝑖
Allowed substitution hints:   𝑁(𝑖)   𝑋(𝑖)   𝑌(𝑖)

Proof of Theorem 2ffzoeq
StepHypRef Expression
1 eqeq1 2824 . . . . . . . . . . . 12 (𝐹 = 𝑃 → (𝐹 = ∅ ↔ 𝑃 = ∅))
21anbi1d 631 . . . . . . . . . . 11 (𝐹 = 𝑃 → ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
3 f0bi 6538 . . . . . . . . . . . . 13 (𝑃:∅⟶𝑌𝑃 = ∅)
4 ffn 6490 . . . . . . . . . . . . . 14 (𝑃:(0..^𝑁)⟶𝑌𝑃 Fn (0..^𝑁))
5 ffn 6490 . . . . . . . . . . . . . 14 (𝑃:∅⟶𝑌𝑃 Fn ∅)
6 fndmu 6434 . . . . . . . . . . . . . . . 16 ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → (0..^𝑁) = ∅)
7 0z 11971 . . . . . . . . . . . . . . . . . 18 0 ∈ ℤ
8 nn0z 11984 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
98adantl 484 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝑁 ∈ ℤ)
10 fzon 13042 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
117, 9, 10sylancr 589 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
12 nn0ge0 11901 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
13 0red 10622 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 ∈ ℝ)
14 nn0re 11885 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
1513, 14letri3d 10760 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (0 = 𝑁 ↔ (0 ≤ 𝑁𝑁 ≤ 0)))
1615biimprd 250 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → ((0 ≤ 𝑁𝑁 ≤ 0) → 0 = 𝑁))
1712, 16mpand 693 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (𝑁 ≤ 0 → 0 = 𝑁))
1817adantl 484 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑁 ≤ 0 → 0 = 𝑁))
1911, 18sylbird 262 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ((0..^𝑁) = ∅ → 0 = 𝑁))
206, 19syl5com 31 . . . . . . . . . . . . . . 15 ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 = 𝑁))
2120ex 415 . . . . . . . . . . . . . 14 (𝑃 Fn (0..^𝑁) → (𝑃 Fn ∅ → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 = 𝑁)))
224, 5, 21syl2imc 41 . . . . . . . . . . . . 13 (𝑃:∅⟶𝑌 → (𝑃:(0..^𝑁)⟶𝑌 → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 = 𝑁)))
233, 22sylbir 237 . . . . . . . . . . . 12 (𝑃 = ∅ → (𝑃:(0..^𝑁)⟶𝑌 → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 = 𝑁)))
2423imp 409 . . . . . . . . . . 11 ((𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 = 𝑁))
252, 24syl6bi 255 . . . . . . . . . 10 (𝐹 = 𝑃 → ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 = 𝑁)))
2625com3l 89 . . . . . . . . 9 ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐹 = 𝑃 → 0 = 𝑁)))
2726a1i 11 . . . . . . . 8 (𝑀 = 0 → ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐹 = 𝑃 → 0 = 𝑁))))
28 oveq2 7141 . . . . . . . . . . . 12 (𝑀 = 0 → (0..^𝑀) = (0..^0))
29 fzo0 13045 . . . . . . . . . . . 12 (0..^0) = ∅
3028, 29syl6eq 2871 . . . . . . . . . . 11 (𝑀 = 0 → (0..^𝑀) = ∅)
3130feq2d 6476 . . . . . . . . . 10 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹:∅⟶𝑋))
32 f0bi 6538 . . . . . . . . . 10 (𝐹:∅⟶𝑋𝐹 = ∅)
3331, 32syl6bb 289 . . . . . . . . 9 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹 = ∅))
3433anbi1d 631 . . . . . . . 8 (𝑀 = 0 → ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) ↔ (𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
35 eqeq1 2824 . . . . . . . . . 10 (𝑀 = 0 → (𝑀 = 𝑁 ↔ 0 = 𝑁))
3635imbi2d 343 . . . . . . . . 9 (𝑀 = 0 → ((𝐹 = 𝑃𝑀 = 𝑁) ↔ (𝐹 = 𝑃 → 0 = 𝑁)))
3736imbi2d 343 . . . . . . . 8 (𝑀 = 0 → (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐹 = 𝑃𝑀 = 𝑁)) ↔ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐹 = 𝑃 → 0 = 𝑁))))
3827, 34, 373imtr4d 296 . . . . . . 7 (𝑀 = 0 → ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐹 = 𝑃𝑀 = 𝑁))))
3938com3l 89 . . . . . 6 ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 = 0 → (𝐹 = 𝑃𝑀 = 𝑁))))
4039impcom 410 . . . . 5 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝐹 = 𝑃𝑀 = 𝑁)))
4140impcom 410 . . . 4 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃𝑀 = 𝑁))
4228feq2d 6476 . . . . . . . . . . . 12 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹:(0..^0)⟶𝑋))
4329feq2i 6482 . . . . . . . . . . . . 13 (𝐹:(0..^0)⟶𝑋𝐹:∅⟶𝑋)
4443, 32bitri 277 . . . . . . . . . . . 12 (𝐹:(0..^0)⟶𝑋𝐹 = ∅)
4542, 44syl6bb 289 . . . . . . . . . . 11 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹 = ∅))
4645adantr 483 . . . . . . . . . 10 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → (𝐹:(0..^𝑀)⟶𝑋𝐹 = ∅))
47 eqeq1 2824 . . . . . . . . . . . 12 (𝑀 = 𝑁 → (𝑀 = 0 ↔ 𝑁 = 0))
4847biimpac 481 . . . . . . . . . . 11 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → 𝑁 = 0)
49 oveq2 7141 . . . . . . . . . . . . 13 (𝑁 = 0 → (0..^𝑁) = (0..^0))
5049feq2d 6476 . . . . . . . . . . . 12 (𝑁 = 0 → (𝑃:(0..^𝑁)⟶𝑌𝑃:(0..^0)⟶𝑌))
5129feq2i 6482 . . . . . . . . . . . . 13 (𝑃:(0..^0)⟶𝑌𝑃:∅⟶𝑌)
5251, 3bitri 277 . . . . . . . . . . . 12 (𝑃:(0..^0)⟶𝑌𝑃 = ∅)
5350, 52syl6bb 289 . . . . . . . . . . 11 (𝑁 = 0 → (𝑃:(0..^𝑁)⟶𝑌𝑃 = ∅))
5448, 53syl 17 . . . . . . . . . 10 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → (𝑃:(0..^𝑁)⟶𝑌𝑃 = ∅))
5546, 54anbi12d 632 . . . . . . . . 9 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) ↔ (𝐹 = ∅ ∧ 𝑃 = ∅)))
56 eqtr3 2842 . . . . . . . . 9 ((𝐹 = ∅ ∧ 𝑃 = ∅) → 𝐹 = 𝑃)
5755, 56syl6bi 255 . . . . . . . 8 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) → 𝐹 = 𝑃))
5857com12 32 . . . . . . 7 ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 = 0 ∧ 𝑀 = 𝑁) → 𝐹 = 𝑃))
5958expd 418 . . . . . 6 ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) → (𝑀 = 0 → (𝑀 = 𝑁𝐹 = 𝑃)))
6059adantl 484 . . . . 5 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝑀 = 𝑁𝐹 = 𝑃)))
6160impcom 410 . . . 4 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝑀 = 𝑁𝐹 = 𝑃))
6241, 61impbid 214 . . 3 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃𝑀 = 𝑁))
63 ral0 4432 . . . . . 6 𝑖 ∈ ∅ (𝐹𝑖) = (𝑃𝑖)
6430raleqdv 3398 . . . . . 6 (𝑀 = 0 → (∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖) ↔ ∀𝑖 ∈ ∅ (𝐹𝑖) = (𝑃𝑖)))
6563, 64mpbiri 260 . . . . 5 (𝑀 = 0 → ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))
6665biantrud 534 . . . 4 (𝑀 = 0 → (𝑀 = 𝑁 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
6766adantr 483 . . 3 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝑀 = 𝑁 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
6862, 67bitrd 281 . 2 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
69 ffn 6490 . . . . . . 7 (𝐹:(0..^𝑀)⟶𝑋𝐹 Fn (0..^𝑀))
7069, 4anim12i 614 . . . . . 6 ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
7170adantl 484 . . . . 5 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
7271adantl 484 . . . 4 ((¬ 𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
73 eqfnfv2 6779 . . . 4 ((𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)) → (𝐹 = 𝑃 ↔ ((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
7472, 73syl 17 . . 3 ((¬ 𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 ↔ ((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
75 df-ne 3007 . . . . . 6 (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
76 elnnne0 11890 . . . . . . . 8 (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℕ0𝑀 ≠ 0))
77 0zd 11972 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → 0 ∈ ℤ)
78 nnz 11983 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
79 nngt0 11647 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → 0 < 𝑀)
8077, 78, 793jca 1124 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
8180adantr 483 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
82 fzoopth 43675 . . . . . . . . . . . . 13 ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀) → ((0..^𝑀) = (0..^𝑁) ↔ (0 = 0 ∧ 𝑀 = 𝑁)))
8381, 82syl 17 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → ((0..^𝑀) = (0..^𝑁) ↔ (0 = 0 ∧ 𝑀 = 𝑁)))
84 simpr 487 . . . . . . . . . . . 12 ((0 = 0 ∧ 𝑀 = 𝑁) → 𝑀 = 𝑁)
8583, 84syl6bi 255 . . . . . . . . . . 11 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → ((0..^𝑀) = (0..^𝑁) → 𝑀 = 𝑁))
8685anim1d 612 . . . . . . . . . 10 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) → (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
87 oveq2 7141 . . . . . . . . . . 11 (𝑀 = 𝑁 → (0..^𝑀) = (0..^𝑁))
8887anim1i 616 . . . . . . . . . 10 ((𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) → ((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))
8986, 88impbid1 227 . . . . . . . . 9 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
9089ex 415 . . . . . . . 8 (𝑀 ∈ ℕ → (𝑁 ∈ ℕ0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9176, 90sylbir 237 . . . . . . 7 ((𝑀 ∈ ℕ0𝑀 ≠ 0) → (𝑁 ∈ ℕ0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9291impancom 454 . . . . . 6 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 ≠ 0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9375, 92syl5bir 245 . . . . 5 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (¬ 𝑀 = 0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9493adantr 483 . . . 4 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (¬ 𝑀 = 0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9594impcom 410 . . 3 ((¬ 𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
9674, 95bitrd 281 . 2 ((¬ 𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
9768, 96pm2.61ian 810 1 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wne 3006  wral 3125  c0 4269   class class class wbr 5042   Fn wfn 6326  wf 6327  cfv 6331  (class class class)co 7133  0cc0 10515   < clt 10653  cle 10654  cn 11616  0cn0 11876  cz 11960  ..^cfzo 13017
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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-er 8267  df-en 8488  df-dom 8489  df-sdom 8490  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-nn 11617  df-n0 11877  df-z 11961  df-uz 12223  df-fz 12877  df-fzo 13018
This theorem is referenced by: (None)
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