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Theorem 2ffzoeq 41817
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 2752 . . . . . . . . . . . 12 (𝐹 = 𝑃 → (𝐹 = ∅ ↔ 𝑃 = ∅))
21anbi1d 743 . . . . . . . . . . 11 (𝐹 = 𝑃 → ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
3 f0bi 6237 . . . . . . . . . . . . 13 (𝑃:∅⟶𝑌𝑃 = ∅)
4 ffn 6194 . . . . . . . . . . . . . 14 (𝑃:(0..^𝑁)⟶𝑌𝑃 Fn (0..^𝑁))
5 ffn 6194 . . . . . . . . . . . . . 14 (𝑃:∅⟶𝑌𝑃 Fn ∅)
6 fndmu 6141 . . . . . . . . . . . . . . . 16 ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → (0..^𝑁) = ∅)
7 0z 11551 . . . . . . . . . . . . . . . . . 18 0 ∈ ℤ
8 nn0z 11563 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
98adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝑁 ∈ ℤ)
10 fzon 12654 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
117, 9, 10sylancr 698 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
12 nn0ge0 11481 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
13 0red 10204 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 ∈ ℝ)
14 nn0re 11464 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
1513, 14letri3d 10342 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (0 = 𝑁 ↔ (0 ≤ 𝑁𝑁 ≤ 0)))
1615biimprd 238 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → ((0 ≤ 𝑁𝑁 ≤ 0) → 0 = 𝑁))
1712, 16mpand 713 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (𝑁 ≤ 0 → 0 = 𝑁))
1817adantl 473 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑁 ≤ 0 → 0 = 𝑁))
1911, 18sylbird 250 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ((0..^𝑁) = ∅ → 0 = 𝑁))
206, 19syl5com 31 . . . . . . . . . . . . . . 15 ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 = 𝑁))
2120ex 449 . . . . . . . . . . . . . 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 225 . . . . . . . . . . . 12 (𝑃 = ∅ → (𝑃:(0..^𝑁)⟶𝑌 → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 = 𝑁)))
2423imp 444 . . . . . . . . . . 11 ((𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 = 𝑁))
252, 24syl6bi 243 . . . . . . . . . 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 6809 . . . . . . . . . . . 12 (𝑀 = 0 → (0..^𝑀) = (0..^0))
29 fzo0 12657 . . . . . . . . . . . 12 (0..^0) = ∅
3028, 29syl6eq 2798 . . . . . . . . . . 11 (𝑀 = 0 → (0..^𝑀) = ∅)
3130feq2d 6180 . . . . . . . . . 10 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹:∅⟶𝑋))
32 f0bi 6237 . . . . . . . . . 10 (𝐹:∅⟶𝑋𝐹 = ∅)
3331, 32syl6bb 276 . . . . . . . . 9 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹 = ∅))
3433anbi1d 743 . . . . . . . 8 (𝑀 = 0 → ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) ↔ (𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
35 eqeq1 2752 . . . . . . . . . 10 (𝑀 = 0 → (𝑀 = 𝑁 ↔ 0 = 𝑁))
3635imbi2d 329 . . . . . . . . 9 (𝑀 = 0 → ((𝐹 = 𝑃𝑀 = 𝑁) ↔ (𝐹 = 𝑃 → 0 = 𝑁)))
3736imbi2d 329 . . . . . . . 8 (𝑀 = 0 → (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐹 = 𝑃𝑀 = 𝑁)) ↔ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐹 = 𝑃 → 0 = 𝑁))))
3827, 34, 373imtr4d 283 . . . . . . 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 445 . . . . 5 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝐹 = 𝑃𝑀 = 𝑁)))
4140impcom 445 . . . 4 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃𝑀 = 𝑁))
4228feq2d 6180 . . . . . . . . . . . 12 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹:(0..^0)⟶𝑋))
4329feq2i 6186 . . . . . . . . . . . . 13 (𝐹:(0..^0)⟶𝑋𝐹:∅⟶𝑋)
4443, 32bitri 264 . . . . . . . . . . . 12 (𝐹:(0..^0)⟶𝑋𝐹 = ∅)
4542, 44syl6bb 276 . . . . . . . . . . 11 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹 = ∅))
4645adantr 472 . . . . . . . . . 10 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → (𝐹:(0..^𝑀)⟶𝑋𝐹 = ∅))
47 eqeq1 2752 . . . . . . . . . . . 12 (𝑀 = 𝑁 → (𝑀 = 0 ↔ 𝑁 = 0))
4847biimpac 504 . . . . . . . . . . 11 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → 𝑁 = 0)
49 oveq2 6809 . . . . . . . . . . . . 13 (𝑁 = 0 → (0..^𝑁) = (0..^0))
5049feq2d 6180 . . . . . . . . . . . 12 (𝑁 = 0 → (𝑃:(0..^𝑁)⟶𝑌𝑃:(0..^0)⟶𝑌))
5129feq2i 6186 . . . . . . . . . . . . 13 (𝑃:(0..^0)⟶𝑌𝑃:∅⟶𝑌)
5251, 3bitri 264 . . . . . . . . . . . 12 (𝑃:(0..^0)⟶𝑌𝑃 = ∅)
5350, 52syl6bb 276 . . . . . . . . . . 11 (𝑁 = 0 → (𝑃:(0..^𝑁)⟶𝑌𝑃 = ∅))
5448, 53syl 17 . . . . . . . . . 10 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → (𝑃:(0..^𝑁)⟶𝑌𝑃 = ∅))
5546, 54anbi12d 749 . . . . . . . . 9 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) ↔ (𝐹 = ∅ ∧ 𝑃 = ∅)))
56 eqtr3 2769 . . . . . . . . 9 ((𝐹 = ∅ ∧ 𝑃 = ∅) → 𝐹 = 𝑃)
5755, 56syl6bi 243 . . . . . . . 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 451 . . . . . 6 ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) → (𝑀 = 0 → (𝑀 = 𝑁𝐹 = 𝑃)))
6059adantl 473 . . . . 5 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝑀 = 𝑁𝐹 = 𝑃)))
6160impcom 445 . . . 4 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝑀 = 𝑁𝐹 = 𝑃))
6241, 61impbid 202 . . 3 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃𝑀 = 𝑁))
63 ral0 4208 . . . . . 6 𝑖 ∈ ∅ (𝐹𝑖) = (𝑃𝑖)
6430raleqdv 3271 . . . . . 6 (𝑀 = 0 → (∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖) ↔ ∀𝑖 ∈ ∅ (𝐹𝑖) = (𝑃𝑖)))
6563, 64mpbiri 248 . . . . 5 (𝑀 = 0 → ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))
6665biantrud 529 . . . 4 (𝑀 = 0 → (𝑀 = 𝑁 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
6766adantr 472 . . 3 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝑀 = 𝑁 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
6862, 67bitrd 268 . 2 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
69 ffn 6194 . . . . . . 7 (𝐹:(0..^𝑀)⟶𝑋𝐹 Fn (0..^𝑀))
7069, 4anim12i 591 . . . . . 6 ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
7170adantl 473 . . . . 5 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
7271adantl 473 . . . 4 ((¬ 𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
73 eqfnfv2 6463 . . . 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 2921 . . . . . 6 (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
76 elnnne0 11469 . . . . . . . 8 (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℕ0𝑀 ≠ 0))
77 0zd 11552 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → 0 ∈ ℤ)
78 nnz 11562 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
79 nngt0 11212 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → 0 < 𝑀)
8077, 78, 793jca 1403 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
8180adantr 472 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
82 fzoopth 41816 . . . . . . . . . . . . 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 479 . . . . . . . . . . . 12 ((0 = 0 ∧ 𝑀 = 𝑁) → 𝑀 = 𝑁)
8583, 84syl6bi 243 . . . . . . . . . . 11 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → ((0..^𝑀) = (0..^𝑁) → 𝑀 = 𝑁))
8685anim1d 589 . . . . . . . . . 10 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) → (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
87 oveq2 6809 . . . . . . . . . . 11 (𝑀 = 𝑁 → (0..^𝑀) = (0..^𝑁))
8887anim1i 593 . . . . . . . . . 10 ((𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) → ((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))
8986, 88impbid1 215 . . . . . . . . 9 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
9089ex 449 . . . . . . . 8 (𝑀 ∈ ℕ → (𝑁 ∈ ℕ0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9176, 90sylbir 225 . . . . . . 7 ((𝑀 ∈ ℕ0𝑀 ≠ 0) → (𝑁 ∈ ℕ0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9291impancom 455 . . . . . 6 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 ≠ 0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9375, 92syl5bir 233 . . . . 5 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (¬ 𝑀 = 0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9493adantr 472 . . . 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 445 . . 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 268 . 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 866 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 196  wa 383  w3a 1072   = wceq 1620  wcel 2127  wne 2920  wral 3038  c0 4046   class class class wbr 4792   Fn wfn 6032  wf 6033  cfv 6037  (class class class)co 6801  0cc0 10099   < clt 10237  cle 10238  cn 11183  0cn0 11455  cz 11540  ..^cfzo 12630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-er 7899  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-nn 11184  df-n0 11456  df-z 11541  df-uz 11851  df-fz 12491  df-fzo 12631
This theorem is referenced by: (None)
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