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Theorem 2ffzoeq 45713
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 2735 . . . . . . . . . . . 12 (𝐹 = 𝑃 → (𝐹 = ∅ ↔ 𝑃 = ∅))
21anbi1d 630 . . . . . . . . . . 11 (𝐹 = 𝑃 → ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
3 f0bi 6745 . . . . . . . . . . . . 13 (𝑃:∅⟶𝑌𝑃 = ∅)
4 ffn 6688 . . . . . . . . . . . . . 14 (𝑃:(0..^𝑁)⟶𝑌𝑃 Fn (0..^𝑁))
5 ffn 6688 . . . . . . . . . . . . . 14 (𝑃:∅⟶𝑌𝑃 Fn ∅)
6 fndmu 6629 . . . . . . . . . . . . . . . 16 ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → (0..^𝑁) = ∅)
7 0z 12534 . . . . . . . . . . . . . . . . . 18 0 ∈ ℤ
8 nn0z 12548 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
98adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝑁 ∈ ℤ)
10 fzon 13618 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
117, 9, 10sylancr 587 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
12 nn0ge0 12462 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
13 0red 11182 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 ∈ ℝ)
14 nn0re 12446 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
1513, 14letri3d 11321 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (0 = 𝑁 ↔ (0 ≤ 𝑁𝑁 ≤ 0)))
1615biimprd 247 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → ((0 ≤ 𝑁𝑁 ≤ 0) → 0 = 𝑁))
1712, 16mpand 693 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (𝑁 ≤ 0 → 0 = 𝑁))
1817adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑁 ≤ 0 → 0 = 𝑁))
1911, 18sylbird 259 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ((0..^𝑁) = ∅ → 0 = 𝑁))
206, 19syl5com 31 . . . . . . . . . . . . . . 15 ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 = 𝑁))
2120ex 413 . . . . . . . . . . . . . 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 234 . . . . . . . . . . . 12 (𝑃 = ∅ → (𝑃:(0..^𝑁)⟶𝑌 → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 = 𝑁)))
2423imp 407 . . . . . . . . . . 11 ((𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 = 𝑁))
252, 24syl6bi 252 . . . . . . . . . 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 7385 . . . . . . . . . . . 12 (𝑀 = 0 → (0..^𝑀) = (0..^0))
29 fzo0 13621 . . . . . . . . . . . 12 (0..^0) = ∅
3028, 29eqtrdi 2787 . . . . . . . . . . 11 (𝑀 = 0 → (0..^𝑀) = ∅)
3130feq2d 6674 . . . . . . . . . 10 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹:∅⟶𝑋))
32 f0bi 6745 . . . . . . . . . 10 (𝐹:∅⟶𝑋𝐹 = ∅)
3331, 32bitrdi 286 . . . . . . . . 9 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹 = ∅))
3433anbi1d 630 . . . . . . . 8 (𝑀 = 0 → ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) ↔ (𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
35 eqeq1 2735 . . . . . . . . . 10 (𝑀 = 0 → (𝑀 = 𝑁 ↔ 0 = 𝑁))
3635imbi2d 340 . . . . . . . . 9 (𝑀 = 0 → ((𝐹 = 𝑃𝑀 = 𝑁) ↔ (𝐹 = 𝑃 → 0 = 𝑁)))
3736imbi2d 340 . . . . . . . 8 (𝑀 = 0 → (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐹 = 𝑃𝑀 = 𝑁)) ↔ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐹 = 𝑃 → 0 = 𝑁))))
3827, 34, 373imtr4d 293 . . . . . . 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 408 . . . . 5 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝐹 = 𝑃𝑀 = 𝑁)))
4140impcom 408 . . . 4 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃𝑀 = 𝑁))
4228feq2d 6674 . . . . . . . . . . . 12 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹:(0..^0)⟶𝑋))
4329feq2i 6680 . . . . . . . . . . . . 13 (𝐹:(0..^0)⟶𝑋𝐹:∅⟶𝑋)
4443, 32bitri 274 . . . . . . . . . . . 12 (𝐹:(0..^0)⟶𝑋𝐹 = ∅)
4542, 44bitrdi 286 . . . . . . . . . . 11 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹 = ∅))
4645adantr 481 . . . . . . . . . 10 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → (𝐹:(0..^𝑀)⟶𝑋𝐹 = ∅))
47 eqeq1 2735 . . . . . . . . . . . 12 (𝑀 = 𝑁 → (𝑀 = 0 ↔ 𝑁 = 0))
4847biimpac 479 . . . . . . . . . . 11 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → 𝑁 = 0)
49 oveq2 7385 . . . . . . . . . . . . 13 (𝑁 = 0 → (0..^𝑁) = (0..^0))
5049feq2d 6674 . . . . . . . . . . . 12 (𝑁 = 0 → (𝑃:(0..^𝑁)⟶𝑌𝑃:(0..^0)⟶𝑌))
5129feq2i 6680 . . . . . . . . . . . . 13 (𝑃:(0..^0)⟶𝑌𝑃:∅⟶𝑌)
5251, 3bitri 274 . . . . . . . . . . . 12 (𝑃:(0..^0)⟶𝑌𝑃 = ∅)
5350, 52bitrdi 286 . . . . . . . . . . 11 (𝑁 = 0 → (𝑃:(0..^𝑁)⟶𝑌𝑃 = ∅))
5448, 53syl 17 . . . . . . . . . 10 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → (𝑃:(0..^𝑁)⟶𝑌𝑃 = ∅))
5546, 54anbi12d 631 . . . . . . . . 9 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) ↔ (𝐹 = ∅ ∧ 𝑃 = ∅)))
56 eqtr3 2757 . . . . . . . . 9 ((𝐹 = ∅ ∧ 𝑃 = ∅) → 𝐹 = 𝑃)
5755, 56syl6bi 252 . . . . . . . 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 416 . . . . . 6 ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) → (𝑀 = 0 → (𝑀 = 𝑁𝐹 = 𝑃)))
6059adantl 482 . . . . 5 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝑀 = 𝑁𝐹 = 𝑃)))
6160impcom 408 . . . 4 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝑀 = 𝑁𝐹 = 𝑃))
6241, 61impbid 211 . . 3 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃𝑀 = 𝑁))
63 ral0 4490 . . . . . 6 𝑖 ∈ ∅ (𝐹𝑖) = (𝑃𝑖)
6430raleqdv 3324 . . . . . 6 (𝑀 = 0 → (∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖) ↔ ∀𝑖 ∈ ∅ (𝐹𝑖) = (𝑃𝑖)))
6563, 64mpbiri 257 . . . . 5 (𝑀 = 0 → ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))
6665biantrud 532 . . . 4 (𝑀 = 0 → (𝑀 = 𝑁 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
6766adantr 481 . . 3 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝑀 = 𝑁 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
6862, 67bitrd 278 . 2 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
69 ffn 6688 . . . . . . 7 (𝐹:(0..^𝑀)⟶𝑋𝐹 Fn (0..^𝑀))
7069, 4anim12i 613 . . . . . 6 ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
7170adantl 482 . . . . 5 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
7271adantl 482 . . . 4 ((¬ 𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
73 eqfnfv2 7003 . . . 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 2940 . . . . . 6 (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
76 elnnne0 12451 . . . . . . . 8 (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℕ0𝑀 ≠ 0))
77 0zd 12535 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → 0 ∈ ℤ)
78 nnz 12544 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
79 nngt0 12208 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → 0 < 𝑀)
8077, 78, 793jca 1128 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
8180adantr 481 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
82 fzoopth 45712 . . . . . . . . . . . . 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 485 . . . . . . . . . . . 12 ((0 = 0 ∧ 𝑀 = 𝑁) → 𝑀 = 𝑁)
8583, 84syl6bi 252 . . . . . . . . . . 11 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → ((0..^𝑀) = (0..^𝑁) → 𝑀 = 𝑁))
8685anim1d 611 . . . . . . . . . 10 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) → (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
87 oveq2 7385 . . . . . . . . . . 11 (𝑀 = 𝑁 → (0..^𝑀) = (0..^𝑁))
8887anim1i 615 . . . . . . . . . 10 ((𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) → ((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))
8986, 88impbid1 224 . . . . . . . . 9 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
9089ex 413 . . . . . . . 8 (𝑀 ∈ ℕ → (𝑁 ∈ ℕ0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9176, 90sylbir 234 . . . . . . 7 ((𝑀 ∈ ℕ0𝑀 ≠ 0) → (𝑁 ∈ ℕ0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9291impancom 452 . . . . . 6 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 ≠ 0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9375, 92biimtrrid 242 . . . . 5 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (¬ 𝑀 = 0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9493adantr 481 . . . 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 408 . . 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 278 . 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2939  wral 3060  c0 4302   class class class wbr 5125   Fn wfn 6511  wf 6512  cfv 6516  (class class class)co 7377  0cc0 11075   < clt 11213  cle 11214  cn 12177  0cn0 12437  cz 12523  ..^cfzo 13592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-1st 7941  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-er 8670  df-en 8906  df-dom 8907  df-sdom 8908  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-nn 12178  df-n0 12438  df-z 12524  df-uz 12788  df-fz 13450  df-fzo 13593
This theorem is referenced by: (None)
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