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Theorem 2ffzoeq 47276
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 2738 . . . . . . . . . . . 12 (𝐹 = 𝑃 → (𝐹 = ∅ ↔ 𝑃 = ∅))
21anbi1d 631 . . . . . . . . . . 11 (𝐹 = 𝑃 → ((𝐹 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) ↔ (𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌)))
3 f0bi 6791 . . . . . . . . . . . . 13 (𝑃:∅⟶𝑌𝑃 = ∅)
4 ffn 6736 . . . . . . . . . . . . . 14 (𝑃:(0..^𝑁)⟶𝑌𝑃 Fn (0..^𝑁))
5 ffn 6736 . . . . . . . . . . . . . 14 (𝑃:∅⟶𝑌𝑃 Fn ∅)
6 fndmu 6675 . . . . . . . . . . . . . . . 16 ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → (0..^𝑁) = ∅)
7 0z 12621 . . . . . . . . . . . . . . . . . 18 0 ∈ ℤ
8 nn0z 12635 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
98adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝑁 ∈ ℤ)
10 fzon 13716 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
117, 9, 10sylancr 587 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑁 ≤ 0 ↔ (0..^𝑁) = ∅))
12 nn0ge0 12548 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
13 0red 11261 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 ∈ ℝ)
14 nn0re 12532 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
1513, 14letri3d 11400 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (0 = 𝑁 ↔ (0 ≤ 𝑁𝑁 ≤ 0)))
1615biimprd 248 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → ((0 ≤ 𝑁𝑁 ≤ 0) → 0 = 𝑁))
1712, 16mpand 695 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (𝑁 ≤ 0 → 0 = 𝑁))
1817adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑁 ≤ 0 → 0 = 𝑁))
1911, 18sylbird 260 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ((0..^𝑁) = ∅ → 0 = 𝑁))
206, 19syl5com 31 . . . . . . . . . . . . . . 15 ((𝑃 Fn (0..^𝑁) ∧ 𝑃 Fn ∅) → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 = 𝑁))
2120ex 412 . . . . . . . . . . . . . 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 235 . . . . . . . . . . . 12 (𝑃 = ∅ → (𝑃:(0..^𝑁)⟶𝑌 → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 = 𝑁)))
2423imp 406 . . . . . . . . . . 11 ((𝑃 = ∅ ∧ 𝑃:(0..^𝑁)⟶𝑌) → ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 0 = 𝑁))
252, 24biimtrdi 253 . . . . . . . . . 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 7438 . . . . . . . . . . . 12 (𝑀 = 0 → (0..^𝑀) = (0..^0))
29 fzo0 13719 . . . . . . . . . . . 12 (0..^0) = ∅
3028, 29eqtrdi 2790 . . . . . . . . . . 11 (𝑀 = 0 → (0..^𝑀) = ∅)
3130feq2d 6722 . . . . . . . . . 10 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹:∅⟶𝑋))
32 f0bi 6791 . . . . . . . . . 10 (𝐹:∅⟶𝑋𝐹 = ∅)
3331, 32bitrdi 287 . . . . . . . . 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 2738 . . . . . . . . . 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 294 . . . . . . 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 407 . . . . 5 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝐹 = 𝑃𝑀 = 𝑁)))
4140impcom 407 . . . 4 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃𝑀 = 𝑁))
4228feq2d 6722 . . . . . . . . . . . 12 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹:(0..^0)⟶𝑋))
4329feq2i 6728 . . . . . . . . . . . . 13 (𝐹:(0..^0)⟶𝑋𝐹:∅⟶𝑋)
4443, 32bitri 275 . . . . . . . . . . . 12 (𝐹:(0..^0)⟶𝑋𝐹 = ∅)
4542, 44bitrdi 287 . . . . . . . . . . 11 (𝑀 = 0 → (𝐹:(0..^𝑀)⟶𝑋𝐹 = ∅))
4645adantr 480 . . . . . . . . . 10 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → (𝐹:(0..^𝑀)⟶𝑋𝐹 = ∅))
47 eqeq1 2738 . . . . . . . . . . . 12 (𝑀 = 𝑁 → (𝑀 = 0 ↔ 𝑁 = 0))
4847biimpac 478 . . . . . . . . . . 11 ((𝑀 = 0 ∧ 𝑀 = 𝑁) → 𝑁 = 0)
49 oveq2 7438 . . . . . . . . . . . . 13 (𝑁 = 0 → (0..^𝑁) = (0..^0))
5049feq2d 6722 . . . . . . . . . . . 12 (𝑁 = 0 → (𝑃:(0..^𝑁)⟶𝑌𝑃:(0..^0)⟶𝑌))
5129feq2i 6728 . . . . . . . . . . . . 13 (𝑃:(0..^0)⟶𝑌𝑃:∅⟶𝑌)
5251, 3bitri 275 . . . . . . . . . . . 12 (𝑃:(0..^0)⟶𝑌𝑃 = ∅)
5350, 52bitrdi 287 . . . . . . . . . . 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 2760 . . . . . . . . 9 ((𝐹 = ∅ ∧ 𝑃 = ∅) → 𝐹 = 𝑃)
5755, 56biimtrdi 253 . . . . . . . 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 415 . . . . . 6 ((𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌) → (𝑀 = 0 → (𝑀 = 𝑁𝐹 = 𝑃)))
6059adantl 481 . . . . 5 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝑀 = 0 → (𝑀 = 𝑁𝐹 = 𝑃)))
6160impcom 407 . . . 4 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝑀 = 𝑁𝐹 = 𝑃))
6241, 61impbid 212 . . 3 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃𝑀 = 𝑁))
63 ral0 4518 . . . . . 6 𝑖 ∈ ∅ (𝐹𝑖) = (𝑃𝑖)
6430raleqdv 3323 . . . . . 6 (𝑀 = 0 → (∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖) ↔ ∀𝑖 ∈ ∅ (𝐹𝑖) = (𝑃𝑖)))
6563, 64mpbiri 258 . . . . 5 (𝑀 = 0 → ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))
6665biantrud 531 . . . 4 (𝑀 = 0 → (𝑀 = 𝑁 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
6766adantr 480 . . 3 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝑀 = 𝑁 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
6862, 67bitrd 279 . 2 ((𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 = 𝑃 ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
69 ffn 6736 . . . . . . 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 481 . . . . 5 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌)) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
7271adantl 481 . . . 4 ((¬ 𝑀 = 0 ∧ ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝐹:(0..^𝑀)⟶𝑋𝑃:(0..^𝑁)⟶𝑌))) → (𝐹 Fn (0..^𝑀) ∧ 𝑃 Fn (0..^𝑁)))
73 eqfnfv2 7051 . . . 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 2938 . . . . . 6 (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
76 elnnne0 12537 . . . . . . . 8 (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℕ0𝑀 ≠ 0))
77 0zd 12622 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → 0 ∈ ℤ)
78 nnz 12631 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
79 nngt0 12294 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → 0 < 𝑀)
8077, 78, 793jca 1127 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
8180adantr 480 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀))
82 fzoopth 13797 . . . . . . . . . . . . 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 484 . . . . . . . . . . . 12 ((0 = 0 ∧ 𝑀 = 𝑁) → 𝑀 = 𝑁)
8583, 84biimtrdi 253 . . . . . . . . . . 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 7438 . . . . . . . . . . 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 225 . . . . . . . . 9 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖))))
9089ex 412 . . . . . . . 8 (𝑀 ∈ ℕ → (𝑁 ∈ ℕ0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9176, 90sylbir 235 . . . . . . 7 ((𝑀 ∈ ℕ0𝑀 ≠ 0) → (𝑁 ∈ ℕ0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9291impancom 451 . . . . . 6 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 ≠ 0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9375, 92biimtrrid 243 . . . . 5 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (¬ 𝑀 = 0 → (((0..^𝑀) = (0..^𝑁) ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝐹𝑖) = (𝑃𝑖)))))
9493adantr 480 . . . 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 407 . . 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 279 . 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 812 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 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  c0 4338   class class class wbr 5147   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  0cc0 11152   < clt 11292  cle 11293  cn 12263  0cn0 12523  cz 12610  ..^cfzo 13690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691
This theorem is referenced by: (None)
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