Step |
Hyp |
Ref |
Expression |
1 |
|
eqeq1 |
⊢ ( 𝐹 = 𝑃 → ( 𝐹 = ∅ ↔ 𝑃 = ∅ ) ) |
2 |
1
|
anbi1d |
⊢ ( 𝐹 = 𝑃 → ( ( 𝐹 = ∅ ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ↔ ( 𝑃 = ∅ ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) ) |
3 |
|
f0bi |
⊢ ( 𝑃 : ∅ ⟶ 𝑌 ↔ 𝑃 = ∅ ) |
4 |
|
ffn |
⊢ ( 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 → 𝑃 Fn ( 0 ..^ 𝑁 ) ) |
5 |
|
ffn |
⊢ ( 𝑃 : ∅ ⟶ 𝑌 → 𝑃 Fn ∅ ) |
6 |
|
fndmu |
⊢ ( ( 𝑃 Fn ( 0 ..^ 𝑁 ) ∧ 𝑃 Fn ∅ ) → ( 0 ..^ 𝑁 ) = ∅ ) |
7 |
|
0z |
⊢ 0 ∈ ℤ |
8 |
|
nn0z |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ ) |
9 |
8
|
adantl |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ℤ ) |
10 |
|
fzon |
⊢ ( ( 0 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 ≤ 0 ↔ ( 0 ..^ 𝑁 ) = ∅ ) ) |
11 |
7 9 10
|
sylancr |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑁 ≤ 0 ↔ ( 0 ..^ 𝑁 ) = ∅ ) ) |
12 |
|
nn0ge0 |
⊢ ( 𝑁 ∈ ℕ0 → 0 ≤ 𝑁 ) |
13 |
|
0red |
⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ ℝ ) |
14 |
|
nn0re |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ ) |
15 |
13 14
|
letri3d |
⊢ ( 𝑁 ∈ ℕ0 → ( 0 = 𝑁 ↔ ( 0 ≤ 𝑁 ∧ 𝑁 ≤ 0 ) ) ) |
16 |
15
|
biimprd |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 0 ≤ 𝑁 ∧ 𝑁 ≤ 0 ) → 0 = 𝑁 ) ) |
17 |
12 16
|
mpand |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 ≤ 0 → 0 = 𝑁 ) ) |
18 |
17
|
adantl |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑁 ≤ 0 → 0 = 𝑁 ) ) |
19 |
11 18
|
sylbird |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( ( 0 ..^ 𝑁 ) = ∅ → 0 = 𝑁 ) ) |
20 |
6 19
|
syl5com |
⊢ ( ( 𝑃 Fn ( 0 ..^ 𝑁 ) ∧ 𝑃 Fn ∅ ) → ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → 0 = 𝑁 ) ) |
21 |
20
|
ex |
⊢ ( 𝑃 Fn ( 0 ..^ 𝑁 ) → ( 𝑃 Fn ∅ → ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → 0 = 𝑁 ) ) ) |
22 |
4 5 21
|
syl2imc |
⊢ ( 𝑃 : ∅ ⟶ 𝑌 → ( 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 → ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → 0 = 𝑁 ) ) ) |
23 |
3 22
|
sylbir |
⊢ ( 𝑃 = ∅ → ( 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 → ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → 0 = 𝑁 ) ) ) |
24 |
23
|
imp |
⊢ ( ( 𝑃 = ∅ ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) → ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → 0 = 𝑁 ) ) |
25 |
2 24
|
syl6bi |
⊢ ( 𝐹 = 𝑃 → ( ( 𝐹 = ∅ ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) → ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → 0 = 𝑁 ) ) ) |
26 |
25
|
com3l |
⊢ ( ( 𝐹 = ∅ ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) → ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝐹 = 𝑃 → 0 = 𝑁 ) ) ) |
27 |
26
|
a1i |
⊢ ( 𝑀 = 0 → ( ( 𝐹 = ∅ ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) → ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝐹 = 𝑃 → 0 = 𝑁 ) ) ) ) |
28 |
|
oveq2 |
⊢ ( 𝑀 = 0 → ( 0 ..^ 𝑀 ) = ( 0 ..^ 0 ) ) |
29 |
|
fzo0 |
⊢ ( 0 ..^ 0 ) = ∅ |
30 |
28 29
|
eqtrdi |
⊢ ( 𝑀 = 0 → ( 0 ..^ 𝑀 ) = ∅ ) |
31 |
30
|
feq2d |
⊢ ( 𝑀 = 0 → ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ↔ 𝐹 : ∅ ⟶ 𝑋 ) ) |
32 |
|
f0bi |
⊢ ( 𝐹 : ∅ ⟶ 𝑋 ↔ 𝐹 = ∅ ) |
33 |
31 32
|
bitrdi |
⊢ ( 𝑀 = 0 → ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ↔ 𝐹 = ∅ ) ) |
34 |
33
|
anbi1d |
⊢ ( 𝑀 = 0 → ( ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ↔ ( 𝐹 = ∅ ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) ) |
35 |
|
eqeq1 |
⊢ ( 𝑀 = 0 → ( 𝑀 = 𝑁 ↔ 0 = 𝑁 ) ) |
36 |
35
|
imbi2d |
⊢ ( 𝑀 = 0 → ( ( 𝐹 = 𝑃 → 𝑀 = 𝑁 ) ↔ ( 𝐹 = 𝑃 → 0 = 𝑁 ) ) ) |
37 |
36
|
imbi2d |
⊢ ( 𝑀 = 0 → ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝐹 = 𝑃 → 𝑀 = 𝑁 ) ) ↔ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝐹 = 𝑃 → 0 = 𝑁 ) ) ) ) |
38 |
27 34 37
|
3imtr4d |
⊢ ( 𝑀 = 0 → ( ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) → ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝐹 = 𝑃 → 𝑀 = 𝑁 ) ) ) ) |
39 |
38
|
com3l |
⊢ ( ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) → ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑀 = 0 → ( 𝐹 = 𝑃 → 𝑀 = 𝑁 ) ) ) ) |
40 |
39
|
impcom |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) → ( 𝑀 = 0 → ( 𝐹 = 𝑃 → 𝑀 = 𝑁 ) ) ) |
41 |
40
|
impcom |
⊢ ( ( 𝑀 = 0 ∧ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) ) → ( 𝐹 = 𝑃 → 𝑀 = 𝑁 ) ) |
42 |
28
|
feq2d |
⊢ ( 𝑀 = 0 → ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ↔ 𝐹 : ( 0 ..^ 0 ) ⟶ 𝑋 ) ) |
43 |
29
|
feq2i |
⊢ ( 𝐹 : ( 0 ..^ 0 ) ⟶ 𝑋 ↔ 𝐹 : ∅ ⟶ 𝑋 ) |
44 |
43 32
|
bitri |
⊢ ( 𝐹 : ( 0 ..^ 0 ) ⟶ 𝑋 ↔ 𝐹 = ∅ ) |
45 |
42 44
|
bitrdi |
⊢ ( 𝑀 = 0 → ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ↔ 𝐹 = ∅ ) ) |
46 |
45
|
adantr |
⊢ ( ( 𝑀 = 0 ∧ 𝑀 = 𝑁 ) → ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ↔ 𝐹 = ∅ ) ) |
47 |
|
eqeq1 |
⊢ ( 𝑀 = 𝑁 → ( 𝑀 = 0 ↔ 𝑁 = 0 ) ) |
48 |
47
|
biimpac |
⊢ ( ( 𝑀 = 0 ∧ 𝑀 = 𝑁 ) → 𝑁 = 0 ) |
49 |
|
oveq2 |
⊢ ( 𝑁 = 0 → ( 0 ..^ 𝑁 ) = ( 0 ..^ 0 ) ) |
50 |
49
|
feq2d |
⊢ ( 𝑁 = 0 → ( 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ↔ 𝑃 : ( 0 ..^ 0 ) ⟶ 𝑌 ) ) |
51 |
29
|
feq2i |
⊢ ( 𝑃 : ( 0 ..^ 0 ) ⟶ 𝑌 ↔ 𝑃 : ∅ ⟶ 𝑌 ) |
52 |
51 3
|
bitri |
⊢ ( 𝑃 : ( 0 ..^ 0 ) ⟶ 𝑌 ↔ 𝑃 = ∅ ) |
53 |
50 52
|
bitrdi |
⊢ ( 𝑁 = 0 → ( 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ↔ 𝑃 = ∅ ) ) |
54 |
48 53
|
syl |
⊢ ( ( 𝑀 = 0 ∧ 𝑀 = 𝑁 ) → ( 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ↔ 𝑃 = ∅ ) ) |
55 |
46 54
|
anbi12d |
⊢ ( ( 𝑀 = 0 ∧ 𝑀 = 𝑁 ) → ( ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ↔ ( 𝐹 = ∅ ∧ 𝑃 = ∅ ) ) ) |
56 |
|
eqtr3 |
⊢ ( ( 𝐹 = ∅ ∧ 𝑃 = ∅ ) → 𝐹 = 𝑃 ) |
57 |
55 56
|
syl6bi |
⊢ ( ( 𝑀 = 0 ∧ 𝑀 = 𝑁 ) → ( ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) → 𝐹 = 𝑃 ) ) |
58 |
57
|
com12 |
⊢ ( ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) → ( ( 𝑀 = 0 ∧ 𝑀 = 𝑁 ) → 𝐹 = 𝑃 ) ) |
59 |
58
|
expd |
⊢ ( ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) → ( 𝑀 = 0 → ( 𝑀 = 𝑁 → 𝐹 = 𝑃 ) ) ) |
60 |
59
|
adantl |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) → ( 𝑀 = 0 → ( 𝑀 = 𝑁 → 𝐹 = 𝑃 ) ) ) |
61 |
60
|
impcom |
⊢ ( ( 𝑀 = 0 ∧ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) ) → ( 𝑀 = 𝑁 → 𝐹 = 𝑃 ) ) |
62 |
41 61
|
impbid |
⊢ ( ( 𝑀 = 0 ∧ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) ) → ( 𝐹 = 𝑃 ↔ 𝑀 = 𝑁 ) ) |
63 |
|
ral0 |
⊢ ∀ 𝑖 ∈ ∅ ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) |
64 |
30
|
raleqdv |
⊢ ( 𝑀 = 0 → ( ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ↔ ∀ 𝑖 ∈ ∅ ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) |
65 |
63 64
|
mpbiri |
⊢ ( 𝑀 = 0 → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) |
66 |
65
|
biantrud |
⊢ ( 𝑀 = 0 → ( 𝑀 = 𝑁 ↔ ( 𝑀 = 𝑁 ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) ) |
67 |
66
|
adantr |
⊢ ( ( 𝑀 = 0 ∧ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) ) → ( 𝑀 = 𝑁 ↔ ( 𝑀 = 𝑁 ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) ) |
68 |
62 67
|
bitrd |
⊢ ( ( 𝑀 = 0 ∧ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) ) → ( 𝐹 = 𝑃 ↔ ( 𝑀 = 𝑁 ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) ) |
69 |
|
ffn |
⊢ ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 → 𝐹 Fn ( 0 ..^ 𝑀 ) ) |
70 |
69 4
|
anim12i |
⊢ ( ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) → ( 𝐹 Fn ( 0 ..^ 𝑀 ) ∧ 𝑃 Fn ( 0 ..^ 𝑁 ) ) ) |
71 |
70
|
adantl |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) → ( 𝐹 Fn ( 0 ..^ 𝑀 ) ∧ 𝑃 Fn ( 0 ..^ 𝑁 ) ) ) |
72 |
71
|
adantl |
⊢ ( ( ¬ 𝑀 = 0 ∧ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) ) → ( 𝐹 Fn ( 0 ..^ 𝑀 ) ∧ 𝑃 Fn ( 0 ..^ 𝑁 ) ) ) |
73 |
|
eqfnfv2 |
⊢ ( ( 𝐹 Fn ( 0 ..^ 𝑀 ) ∧ 𝑃 Fn ( 0 ..^ 𝑁 ) ) → ( 𝐹 = 𝑃 ↔ ( ( 0 ..^ 𝑀 ) = ( 0 ..^ 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) ) |
74 |
72 73
|
syl |
⊢ ( ( ¬ 𝑀 = 0 ∧ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) ) → ( 𝐹 = 𝑃 ↔ ( ( 0 ..^ 𝑀 ) = ( 0 ..^ 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) ) |
75 |
|
df-ne |
⊢ ( 𝑀 ≠ 0 ↔ ¬ 𝑀 = 0 ) |
76 |
|
elnnne0 |
⊢ ( 𝑀 ∈ ℕ ↔ ( 𝑀 ∈ ℕ0 ∧ 𝑀 ≠ 0 ) ) |
77 |
|
0zd |
⊢ ( 𝑀 ∈ ℕ → 0 ∈ ℤ ) |
78 |
|
nnz |
⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℤ ) |
79 |
|
nngt0 |
⊢ ( 𝑀 ∈ ℕ → 0 < 𝑀 ) |
80 |
77 78 79
|
3jca |
⊢ ( 𝑀 ∈ ℕ → ( 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀 ) ) |
81 |
80
|
adantr |
⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0 ) → ( 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀 ) ) |
82 |
|
fzoopth |
⊢ ( ( 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀 ) → ( ( 0 ..^ 𝑀 ) = ( 0 ..^ 𝑁 ) ↔ ( 0 = 0 ∧ 𝑀 = 𝑁 ) ) ) |
83 |
81 82
|
syl |
⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0 ) → ( ( 0 ..^ 𝑀 ) = ( 0 ..^ 𝑁 ) ↔ ( 0 = 0 ∧ 𝑀 = 𝑁 ) ) ) |
84 |
|
simpr |
⊢ ( ( 0 = 0 ∧ 𝑀 = 𝑁 ) → 𝑀 = 𝑁 ) |
85 |
83 84
|
syl6bi |
⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0 ) → ( ( 0 ..^ 𝑀 ) = ( 0 ..^ 𝑁 ) → 𝑀 = 𝑁 ) ) |
86 |
85
|
anim1d |
⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0 ) → ( ( ( 0 ..^ 𝑀 ) = ( 0 ..^ 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) → ( 𝑀 = 𝑁 ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) ) |
87 |
|
oveq2 |
⊢ ( 𝑀 = 𝑁 → ( 0 ..^ 𝑀 ) = ( 0 ..^ 𝑁 ) ) |
88 |
87
|
anim1i |
⊢ ( ( 𝑀 = 𝑁 ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) → ( ( 0 ..^ 𝑀 ) = ( 0 ..^ 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) |
89 |
86 88
|
impbid1 |
⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0 ) → ( ( ( 0 ..^ 𝑀 ) = ( 0 ..^ 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ↔ ( 𝑀 = 𝑁 ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) ) |
90 |
89
|
ex |
⊢ ( 𝑀 ∈ ℕ → ( 𝑁 ∈ ℕ0 → ( ( ( 0 ..^ 𝑀 ) = ( 0 ..^ 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ↔ ( 𝑀 = 𝑁 ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) ) ) |
91 |
76 90
|
sylbir |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑀 ≠ 0 ) → ( 𝑁 ∈ ℕ0 → ( ( ( 0 ..^ 𝑀 ) = ( 0 ..^ 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ↔ ( 𝑀 = 𝑁 ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) ) ) |
92 |
91
|
impancom |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑀 ≠ 0 → ( ( ( 0 ..^ 𝑀 ) = ( 0 ..^ 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ↔ ( 𝑀 = 𝑁 ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) ) ) |
93 |
75 92
|
syl5bir |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( ¬ 𝑀 = 0 → ( ( ( 0 ..^ 𝑀 ) = ( 0 ..^ 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ↔ ( 𝑀 = 𝑁 ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) ) ) |
94 |
93
|
adantr |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) → ( ¬ 𝑀 = 0 → ( ( ( 0 ..^ 𝑀 ) = ( 0 ..^ 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ↔ ( 𝑀 = 𝑁 ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) ) ) |
95 |
94
|
impcom |
⊢ ( ( ¬ 𝑀 = 0 ∧ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) ) → ( ( ( 0 ..^ 𝑀 ) = ( 0 ..^ 𝑁 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ↔ ( 𝑀 = 𝑁 ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) ) |
96 |
74 95
|
bitrd |
⊢ ( ( ¬ 𝑀 = 0 ∧ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) ) → ( 𝐹 = 𝑃 ↔ ( 𝑀 = 𝑁 ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) ) |
97 |
68 96
|
pm2.61ian |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) ∧ ( 𝐹 : ( 0 ..^ 𝑀 ) ⟶ 𝑋 ∧ 𝑃 : ( 0 ..^ 𝑁 ) ⟶ 𝑌 ) ) → ( 𝐹 = 𝑃 ↔ ( 𝑀 = 𝑁 ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐹 ‘ 𝑖 ) = ( 𝑃 ‘ 𝑖 ) ) ) ) |