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Theorem 2ffzoeq 41765
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 2728 . . . . . . . . . . . 12 (𝐹 = 𝑃 → (𝐹 = ∅ ↔ 𝑃 = ∅))
21anbi1d 743 . . . . . . . . . . 11 (𝐹 = 𝑃 → ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
3 f0bi 6201 . . . . . . . . . . . . 13 (𝑃:∅⟶𝑌𝑃 = ∅)
4 ffn 6158 . . . . . . . . . . . . . 14 (𝑃:(0..^𝑁)⟶𝑌𝑃 Fn (0..^𝑁))
5 ffn 6158 . . . . . . . . . . . . . 14 (𝑃:∅⟶𝑌𝑃 Fn ∅)
6 fndmu 6105 . . . . . . . . . . . . . . . 16 ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → (0..^𝑁) = ∅)
7 0z 11501 . . . . . . . . . . . . . . . . . 18 0 ∈ ℤ
8 nn0z 11513 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
98adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝑁 ∈ ℤ)
10 fzon 12604 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
117, 9, 10sylancr 698 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
12 nn0ge0 11431 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
13 0red 10154 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 ∈ ℝ)
14 nn0re 11414 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
1513, 14letri3d 10292 . . . . . . . . . . . . . . . . . . . 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 6773 . . . . . . . . . . . 12 (𝑀 = 0 → (0..^𝑀) = (0..^0))
29 fzo0 12607 . . . . . . . . . . . 12 (0..^0) = ∅
3028, 29syl6eq 2774 . . . . . . . . . . 11 (𝑀 = 0 → (0..^𝑀) = ∅)
3130feq2d 6144 . . . . . . . . . 10 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹:∅⟶𝑋))
32 f0bi 6201 . . . . . . . . . 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 2728 . . . . . . . . . 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 6144 . . . . . . . . . . . 12 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹:(0..^0)⟶𝑋))
4329feq2i 6150 . . . . . . . . . . . . 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 2728 . . . . . . . . . . . 12 (𝑀 = 𝑁 → (𝑀 = 0 ↔ 𝑁 = 0))
4847biimpac 504 . . . . . . . . . . 11 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → 𝑁 = 0)
49 oveq2 6773 . . . . . . . . . . . . 13 (𝑁 = 0 → (0..^𝑁) = (0..^0))
5049feq2d 6144 . . . . . . . . . . . 12 (𝑁 = 0 → (𝑃:(0..^𝑁)⟶𝑌𝑃:(0..^0)⟶𝑌))
5129feq2i 6150 . . . . . . . . . . . . 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 2745 . . . . . . . . 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 4184 . . . . . 6 𝑖 ∈ ∅ (𝐹𝑖) = (𝑃𝑖)
6430raleqdv 3247 . . . . . 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 6158 . . . . . . 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 6427 . . . 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 2897 . . . . . 6 (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
76 elnnne0 11419 . . . . . . . 8 (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℕ0𝑀 ≠ 0))
77 0zd 11502 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → 0 ∈ ℤ)
78 nnz 11512 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
79 nngt0 11162 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → 0 < 𝑀)
8077, 78, 793jca 1379 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
8180adantr 472 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
82 fzoopth 41764 . . . . . . . . . . . . 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 6773 . . . . . . . . . . 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 1596  wcel 2103  wne 2896  wral 3014  c0 4023   class class class wbr 4760   Fn wfn 5996  wf 5997  cfv 6001  (class class class)co 6765  0cc0 10049   < clt 10187  cle 10188  cn 11133  0cn0 11405  cz 11490  ..^cfzo 12580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-n0 11406  df-z 11491  df-uz 11801  df-fz 12441  df-fzo 12581
This theorem is referenced by: (None)
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