Step |
Hyp |
Ref |
Expression |
1 |
|
sticksstones2.1 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
2 |
|
sticksstones2.2 |
⊢ ( 𝜑 → 𝐾 ∈ ℕ0 ) |
3 |
|
sticksstones2.3 |
⊢ 𝐵 = { 𝑎 ∈ 𝒫 ( 1 ... 𝑁 ) ∣ ( ♯ ‘ 𝑎 ) = 𝐾 } |
4 |
|
sticksstones2.4 |
⊢ 𝐴 = { 𝑓 ∣ ( 𝑓 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ) } |
5 |
|
sticksstones2.5 |
⊢ 𝐹 = ( 𝑧 ∈ 𝐴 ↦ ran 𝑧 ) |
6 |
|
fveqeq2 |
⊢ ( 𝑎 = ran 𝑧 → ( ( ♯ ‘ 𝑎 ) = 𝐾 ↔ ( ♯ ‘ ran 𝑧 ) = 𝐾 ) ) |
7 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( 1 ... 𝑁 ) ∈ Fin ) |
8 |
|
eleq1w |
⊢ ( 𝑓 = 𝑧 → ( 𝑓 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴 ) ) |
9 |
|
feq1 |
⊢ ( 𝑓 = 𝑧 → ( 𝑓 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ↔ 𝑧 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ) ) |
10 |
|
fveq1 |
⊢ ( 𝑓 = 𝑧 → ( 𝑓 ‘ 𝑥 ) = ( 𝑧 ‘ 𝑥 ) ) |
11 |
|
fveq1 |
⊢ ( 𝑓 = 𝑧 → ( 𝑓 ‘ 𝑦 ) = ( 𝑧 ‘ 𝑦 ) ) |
12 |
10 11
|
breq12d |
⊢ ( 𝑓 = 𝑧 → ( ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ↔ ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) ) |
13 |
12
|
imbi2d |
⊢ ( 𝑓 = 𝑧 → ( ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ↔ ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) ) ) |
14 |
13
|
ralbidv |
⊢ ( 𝑓 = 𝑧 → ( ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) ) ) |
15 |
14
|
ralbidv |
⊢ ( 𝑓 = 𝑧 → ( ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) ) ) |
16 |
9 15
|
anbi12d |
⊢ ( 𝑓 = 𝑧 → ( ( 𝑓 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ) ↔ ( 𝑧 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) ) ) ) |
17 |
8 16
|
bibi12d |
⊢ ( 𝑓 = 𝑧 → ( ( 𝑓 ∈ 𝐴 ↔ ( 𝑓 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ) ) ↔ ( 𝑧 ∈ 𝐴 ↔ ( 𝑧 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) ) ) ) ) |
18 |
|
abeq2 |
⊢ ( 𝐴 = { 𝑓 ∣ ( 𝑓 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ) } ↔ ∀ 𝑓 ( 𝑓 ∈ 𝐴 ↔ ( 𝑓 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ) ) ) |
19 |
4 18
|
mpbi |
⊢ ∀ 𝑓 ( 𝑓 ∈ 𝐴 ↔ ( 𝑓 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ) ) |
20 |
19
|
spi |
⊢ ( 𝑓 ∈ 𝐴 ↔ ( 𝑓 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ) ) |
21 |
17 20
|
chvarvv |
⊢ ( 𝑧 ∈ 𝐴 ↔ ( 𝑧 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) ) ) |
22 |
21
|
biimpi |
⊢ ( 𝑧 ∈ 𝐴 → ( 𝑧 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) ) ) |
23 |
22
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( 𝑧 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) ) ) |
24 |
23
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝑧 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ) |
25 |
24
|
frnd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ran 𝑧 ⊆ ( 1 ... 𝑁 ) ) |
26 |
7 25
|
sselpwd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ran 𝑧 ∈ 𝒫 ( 1 ... 𝑁 ) ) |
27 |
24
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝑧 Fn ( 1 ... 𝐾 ) ) |
28 |
|
hashfn |
⊢ ( 𝑧 Fn ( 1 ... 𝐾 ) → ( ♯ ‘ 𝑧 ) = ( ♯ ‘ ( 1 ... 𝐾 ) ) ) |
29 |
27 28
|
syl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( ♯ ‘ 𝑧 ) = ( ♯ ‘ ( 1 ... 𝐾 ) ) ) |
30 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝐾 ∈ ℕ0 ) |
31 |
|
hashfz1 |
⊢ ( 𝐾 ∈ ℕ0 → ( ♯ ‘ ( 1 ... 𝐾 ) ) = 𝐾 ) |
32 |
30 31
|
syl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( ♯ ‘ ( 1 ... 𝐾 ) ) = 𝐾 ) |
33 |
29 32
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( ♯ ‘ 𝑧 ) = 𝐾 ) |
34 |
33
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝐾 = ( ♯ ‘ 𝑧 ) ) |
35 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( 1 ... 𝐾 ) ∈ Fin ) |
36 |
|
elfznn |
⊢ ( 𝑎 ∈ ( 1 ... 𝐾 ) → 𝑎 ∈ ℕ ) |
37 |
36
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) → 𝑎 ∈ ℕ ) |
38 |
37
|
nnred |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) → 𝑎 ∈ ℝ ) |
39 |
38
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → 𝑎 ∈ ℝ ) |
40 |
|
elfznn |
⊢ ( 𝑏 ∈ ( 1 ... 𝐾 ) → 𝑏 ∈ ℕ ) |
41 |
40
|
nnred |
⊢ ( 𝑏 ∈ ( 1 ... 𝐾 ) → 𝑏 ∈ ℝ ) |
42 |
41
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → 𝑏 ∈ ℝ ) |
43 |
|
lttri2 |
⊢ ( ( 𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ) → ( 𝑎 ≠ 𝑏 ↔ ( 𝑎 < 𝑏 ∨ 𝑏 < 𝑎 ) ) ) |
44 |
39 42 43
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → ( 𝑎 ≠ 𝑏 ↔ ( 𝑎 < 𝑏 ∨ 𝑏 < 𝑎 ) ) ) |
45 |
24
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) → 𝑧 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ) |
46 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) → 𝑎 ∈ ( 1 ... 𝐾 ) ) |
47 |
45 46
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) → ( 𝑧 ‘ 𝑎 ) ∈ ( 1 ... 𝑁 ) ) |
48 |
47
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → ( 𝑧 ‘ 𝑎 ) ∈ ( 1 ... 𝑁 ) ) |
49 |
48
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑎 < 𝑏 ) → ( 𝑧 ‘ 𝑎 ) ∈ ( 1 ... 𝑁 ) ) |
50 |
|
elfznn |
⊢ ( ( 𝑧 ‘ 𝑎 ) ∈ ( 1 ... 𝑁 ) → ( 𝑧 ‘ 𝑎 ) ∈ ℕ ) |
51 |
49 50
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑎 < 𝑏 ) → ( 𝑧 ‘ 𝑎 ) ∈ ℕ ) |
52 |
51
|
nnred |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑎 < 𝑏 ) → ( 𝑧 ‘ 𝑎 ) ∈ ℝ ) |
53 |
23
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) ) |
54 |
53
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) → ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) ) |
55 |
54
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) ) |
56 |
46
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → 𝑎 ∈ ( 1 ... 𝐾 ) ) |
57 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → 𝑏 ∈ ( 1 ... 𝐾 ) ) |
58 |
|
breq1 |
⊢ ( 𝑥 = 𝑎 → ( 𝑥 < 𝑦 ↔ 𝑎 < 𝑦 ) ) |
59 |
|
fveq2 |
⊢ ( 𝑥 = 𝑎 → ( 𝑧 ‘ 𝑥 ) = ( 𝑧 ‘ 𝑎 ) ) |
60 |
59
|
breq1d |
⊢ ( 𝑥 = 𝑎 → ( ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ↔ ( 𝑧 ‘ 𝑎 ) < ( 𝑧 ‘ 𝑦 ) ) ) |
61 |
58 60
|
imbi12d |
⊢ ( 𝑥 = 𝑎 → ( ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) ↔ ( 𝑎 < 𝑦 → ( 𝑧 ‘ 𝑎 ) < ( 𝑧 ‘ 𝑦 ) ) ) ) |
62 |
|
breq2 |
⊢ ( 𝑦 = 𝑏 → ( 𝑎 < 𝑦 ↔ 𝑎 < 𝑏 ) ) |
63 |
|
fveq2 |
⊢ ( 𝑦 = 𝑏 → ( 𝑧 ‘ 𝑦 ) = ( 𝑧 ‘ 𝑏 ) ) |
64 |
63
|
breq2d |
⊢ ( 𝑦 = 𝑏 → ( ( 𝑧 ‘ 𝑎 ) < ( 𝑧 ‘ 𝑦 ) ↔ ( 𝑧 ‘ 𝑎 ) < ( 𝑧 ‘ 𝑏 ) ) ) |
65 |
62 64
|
imbi12d |
⊢ ( 𝑦 = 𝑏 → ( ( 𝑎 < 𝑦 → ( 𝑧 ‘ 𝑎 ) < ( 𝑧 ‘ 𝑦 ) ) ↔ ( 𝑎 < 𝑏 → ( 𝑧 ‘ 𝑎 ) < ( 𝑧 ‘ 𝑏 ) ) ) ) |
66 |
61 65
|
rspc2v |
⊢ ( ( 𝑎 ∈ ( 1 ... 𝐾 ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → ( ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) → ( 𝑎 < 𝑏 → ( 𝑧 ‘ 𝑎 ) < ( 𝑧 ‘ 𝑏 ) ) ) ) |
67 |
56 57 66
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → ( ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) → ( 𝑎 < 𝑏 → ( 𝑧 ‘ 𝑎 ) < ( 𝑧 ‘ 𝑏 ) ) ) ) |
68 |
55 67
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → ( 𝑎 < 𝑏 → ( 𝑧 ‘ 𝑎 ) < ( 𝑧 ‘ 𝑏 ) ) ) |
69 |
68
|
imp |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑎 < 𝑏 ) → ( 𝑧 ‘ 𝑎 ) < ( 𝑧 ‘ 𝑏 ) ) |
70 |
52 69
|
ltned |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑎 < 𝑏 ) → ( 𝑧 ‘ 𝑎 ) ≠ ( 𝑧 ‘ 𝑏 ) ) |
71 |
45
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → ( 𝑧 ‘ 𝑏 ) ∈ ( 1 ... 𝑁 ) ) |
72 |
|
elfznn |
⊢ ( ( 𝑧 ‘ 𝑏 ) ∈ ( 1 ... 𝑁 ) → ( 𝑧 ‘ 𝑏 ) ∈ ℕ ) |
73 |
71 72
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → ( 𝑧 ‘ 𝑏 ) ∈ ℕ ) |
74 |
73
|
nnred |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → ( 𝑧 ‘ 𝑏 ) ∈ ℝ ) |
75 |
74
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 < 𝑎 ) → ( 𝑧 ‘ 𝑏 ) ∈ ℝ ) |
76 |
|
breq1 |
⊢ ( 𝑥 = 𝑏 → ( 𝑥 < 𝑦 ↔ 𝑏 < 𝑦 ) ) |
77 |
|
fveq2 |
⊢ ( 𝑥 = 𝑏 → ( 𝑧 ‘ 𝑥 ) = ( 𝑧 ‘ 𝑏 ) ) |
78 |
77
|
breq1d |
⊢ ( 𝑥 = 𝑏 → ( ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ↔ ( 𝑧 ‘ 𝑏 ) < ( 𝑧 ‘ 𝑦 ) ) ) |
79 |
76 78
|
imbi12d |
⊢ ( 𝑥 = 𝑏 → ( ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) ↔ ( 𝑏 < 𝑦 → ( 𝑧 ‘ 𝑏 ) < ( 𝑧 ‘ 𝑦 ) ) ) ) |
80 |
|
breq2 |
⊢ ( 𝑦 = 𝑎 → ( 𝑏 < 𝑦 ↔ 𝑏 < 𝑎 ) ) |
81 |
|
fveq2 |
⊢ ( 𝑦 = 𝑎 → ( 𝑧 ‘ 𝑦 ) = ( 𝑧 ‘ 𝑎 ) ) |
82 |
81
|
breq2d |
⊢ ( 𝑦 = 𝑎 → ( ( 𝑧 ‘ 𝑏 ) < ( 𝑧 ‘ 𝑦 ) ↔ ( 𝑧 ‘ 𝑏 ) < ( 𝑧 ‘ 𝑎 ) ) ) |
83 |
80 82
|
imbi12d |
⊢ ( 𝑦 = 𝑎 → ( ( 𝑏 < 𝑦 → ( 𝑧 ‘ 𝑏 ) < ( 𝑧 ‘ 𝑦 ) ) ↔ ( 𝑏 < 𝑎 → ( 𝑧 ‘ 𝑏 ) < ( 𝑧 ‘ 𝑎 ) ) ) ) |
84 |
79 83
|
rspc2v |
⊢ ( ( 𝑏 ∈ ( 1 ... 𝐾 ) ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) → ( ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) → ( 𝑏 < 𝑎 → ( 𝑧 ‘ 𝑏 ) < ( 𝑧 ‘ 𝑎 ) ) ) ) |
85 |
57 56 84
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → ( ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑧 ‘ 𝑥 ) < ( 𝑧 ‘ 𝑦 ) ) → ( 𝑏 < 𝑎 → ( 𝑧 ‘ 𝑏 ) < ( 𝑧 ‘ 𝑎 ) ) ) ) |
86 |
55 85
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → ( 𝑏 < 𝑎 → ( 𝑧 ‘ 𝑏 ) < ( 𝑧 ‘ 𝑎 ) ) ) |
87 |
86
|
imp |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 < 𝑎 ) → ( 𝑧 ‘ 𝑏 ) < ( 𝑧 ‘ 𝑎 ) ) |
88 |
75 87
|
ltned |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 < 𝑎 ) → ( 𝑧 ‘ 𝑏 ) ≠ ( 𝑧 ‘ 𝑎 ) ) |
89 |
88
|
necomd |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 < 𝑎 ) → ( 𝑧 ‘ 𝑎 ) ≠ ( 𝑧 ‘ 𝑏 ) ) |
90 |
70 89
|
jaodan |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) ∧ ( 𝑎 < 𝑏 ∨ 𝑏 < 𝑎 ) ) → ( 𝑧 ‘ 𝑎 ) ≠ ( 𝑧 ‘ 𝑏 ) ) |
91 |
90
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → ( ( 𝑎 < 𝑏 ∨ 𝑏 < 𝑎 ) → ( 𝑧 ‘ 𝑎 ) ≠ ( 𝑧 ‘ 𝑏 ) ) ) |
92 |
44 91
|
sylbid |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → ( 𝑎 ≠ 𝑏 → ( 𝑧 ‘ 𝑎 ) ≠ ( 𝑧 ‘ 𝑏 ) ) ) |
93 |
92
|
necon4d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) ∧ 𝑏 ∈ ( 1 ... 𝐾 ) ) → ( ( 𝑧 ‘ 𝑎 ) = ( 𝑧 ‘ 𝑏 ) → 𝑎 = 𝑏 ) ) |
94 |
93
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) → ∀ 𝑏 ∈ ( 1 ... 𝐾 ) ( ( 𝑧 ‘ 𝑎 ) = ( 𝑧 ‘ 𝑏 ) → 𝑎 = 𝑏 ) ) |
95 |
94
|
3expa |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) ∧ 𝑎 ∈ ( 1 ... 𝐾 ) ) → ∀ 𝑏 ∈ ( 1 ... 𝐾 ) ( ( 𝑧 ‘ 𝑎 ) = ( 𝑧 ‘ 𝑏 ) → 𝑎 = 𝑏 ) ) |
96 |
95
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ∀ 𝑎 ∈ ( 1 ... 𝐾 ) ∀ 𝑏 ∈ ( 1 ... 𝐾 ) ( ( 𝑧 ‘ 𝑎 ) = ( 𝑧 ‘ 𝑏 ) → 𝑎 = 𝑏 ) ) |
97 |
24 96
|
jca |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( 𝑧 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑎 ∈ ( 1 ... 𝐾 ) ∀ 𝑏 ∈ ( 1 ... 𝐾 ) ( ( 𝑧 ‘ 𝑎 ) = ( 𝑧 ‘ 𝑏 ) → 𝑎 = 𝑏 ) ) ) |
98 |
|
dff13 |
⊢ ( 𝑧 : ( 1 ... 𝐾 ) –1-1→ ( 1 ... 𝑁 ) ↔ ( 𝑧 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑎 ∈ ( 1 ... 𝐾 ) ∀ 𝑏 ∈ ( 1 ... 𝐾 ) ( ( 𝑧 ‘ 𝑎 ) = ( 𝑧 ‘ 𝑏 ) → 𝑎 = 𝑏 ) ) ) |
99 |
97 98
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝑧 : ( 1 ... 𝐾 ) –1-1→ ( 1 ... 𝑁 ) ) |
100 |
|
hashf1rn |
⊢ ( ( ( 1 ... 𝐾 ) ∈ Fin ∧ 𝑧 : ( 1 ... 𝐾 ) –1-1→ ( 1 ... 𝑁 ) ) → ( ♯ ‘ 𝑧 ) = ( ♯ ‘ ran 𝑧 ) ) |
101 |
35 99 100
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( ♯ ‘ 𝑧 ) = ( ♯ ‘ ran 𝑧 ) ) |
102 |
34 101
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → 𝐾 = ( ♯ ‘ ran 𝑧 ) ) |
103 |
102
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( ♯ ‘ ran 𝑧 ) = 𝐾 ) |
104 |
6 26 103
|
elrabd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ran 𝑧 ∈ { 𝑎 ∈ 𝒫 ( 1 ... 𝑁 ) ∣ ( ♯ ‘ 𝑎 ) = 𝐾 } ) |
105 |
3
|
eleq2i |
⊢ ( ran 𝑧 ∈ 𝐵 ↔ ran 𝑧 ∈ { 𝑎 ∈ 𝒫 ( 1 ... 𝑁 ) ∣ ( ♯ ‘ 𝑎 ) = 𝐾 } ) |
106 |
105
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ( ran 𝑧 ∈ 𝐵 ↔ ran 𝑧 ∈ { 𝑎 ∈ 𝒫 ( 1 ... 𝑁 ) ∣ ( ♯ ‘ 𝑎 ) = 𝐾 } ) ) |
107 |
104 106
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝐴 ) → ran 𝑧 ∈ 𝐵 ) |
108 |
107 5
|
fmptd |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
109 |
1
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) → 𝑁 ∈ ℕ0 ) |
110 |
109
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) → 𝑁 ∈ ℕ0 ) |
111 |
2
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) → 𝐾 ∈ ℕ0 ) |
112 |
111
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) → 𝐾 ∈ ℕ0 ) |
113 |
|
simpl2 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) → 𝑖 ∈ 𝐴 ) |
114 |
|
simpl3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) → 𝑗 ∈ 𝐴 ) |
115 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) → 𝑖 ≠ 𝑗 ) |
116 |
|
fveq2 |
⊢ ( 𝑟 = 𝑠 → ( 𝑖 ‘ 𝑟 ) = ( 𝑖 ‘ 𝑠 ) ) |
117 |
|
fveq2 |
⊢ ( 𝑟 = 𝑠 → ( 𝑗 ‘ 𝑟 ) = ( 𝑗 ‘ 𝑠 ) ) |
118 |
116 117
|
neeq12d |
⊢ ( 𝑟 = 𝑠 → ( ( 𝑖 ‘ 𝑟 ) ≠ ( 𝑗 ‘ 𝑟 ) ↔ ( 𝑖 ‘ 𝑠 ) ≠ ( 𝑗 ‘ 𝑠 ) ) ) |
119 |
118
|
cbvrabv |
⊢ { 𝑟 ∈ ( 1 ... 𝐾 ) ∣ ( 𝑖 ‘ 𝑟 ) ≠ ( 𝑗 ‘ 𝑟 ) } = { 𝑠 ∈ ( 1 ... 𝐾 ) ∣ ( 𝑖 ‘ 𝑠 ) ≠ ( 𝑗 ‘ 𝑠 ) } |
120 |
119
|
infeq1i |
⊢ inf ( { 𝑟 ∈ ( 1 ... 𝐾 ) ∣ ( 𝑖 ‘ 𝑟 ) ≠ ( 𝑗 ‘ 𝑟 ) } , ℝ , < ) = inf ( { 𝑠 ∈ ( 1 ... 𝐾 ) ∣ ( 𝑖 ‘ 𝑠 ) ≠ ( 𝑗 ‘ 𝑠 ) } , ℝ , < ) |
121 |
110 112 4 113 114 115 120
|
sticksstones1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) → ran 𝑖 ≠ ran 𝑗 ) |
122 |
5
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) → 𝐹 = ( 𝑧 ∈ 𝐴 ↦ ran 𝑧 ) ) |
123 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) ∧ 𝑧 = 𝑖 ) → 𝑧 = 𝑖 ) |
124 |
123
|
rneqd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) ∧ 𝑧 = 𝑖 ) → ran 𝑧 = ran 𝑖 ) |
125 |
|
fzfid |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) → ( 1 ... 𝑁 ) ∈ Fin ) |
126 |
|
eleq1w |
⊢ ( 𝑓 = 𝑖 → ( 𝑓 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴 ) ) |
127 |
|
feq1 |
⊢ ( 𝑓 = 𝑖 → ( 𝑓 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ↔ 𝑖 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ) ) |
128 |
|
fveq1 |
⊢ ( 𝑓 = 𝑖 → ( 𝑓 ‘ 𝑥 ) = ( 𝑖 ‘ 𝑥 ) ) |
129 |
|
fveq1 |
⊢ ( 𝑓 = 𝑖 → ( 𝑓 ‘ 𝑦 ) = ( 𝑖 ‘ 𝑦 ) ) |
130 |
128 129
|
breq12d |
⊢ ( 𝑓 = 𝑖 → ( ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ↔ ( 𝑖 ‘ 𝑥 ) < ( 𝑖 ‘ 𝑦 ) ) ) |
131 |
130
|
imbi2d |
⊢ ( 𝑓 = 𝑖 → ( ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ↔ ( 𝑥 < 𝑦 → ( 𝑖 ‘ 𝑥 ) < ( 𝑖 ‘ 𝑦 ) ) ) ) |
132 |
131
|
2ralbidv |
⊢ ( 𝑓 = 𝑖 → ( ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑖 ‘ 𝑥 ) < ( 𝑖 ‘ 𝑦 ) ) ) ) |
133 |
127 132
|
anbi12d |
⊢ ( 𝑓 = 𝑖 → ( ( 𝑓 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ) ↔ ( 𝑖 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑖 ‘ 𝑥 ) < ( 𝑖 ‘ 𝑦 ) ) ) ) ) |
134 |
126 133
|
bibi12d |
⊢ ( 𝑓 = 𝑖 → ( ( 𝑓 ∈ 𝐴 ↔ ( 𝑓 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ) ) ↔ ( 𝑖 ∈ 𝐴 ↔ ( 𝑖 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑖 ‘ 𝑥 ) < ( 𝑖 ‘ 𝑦 ) ) ) ) ) ) |
135 |
134 20
|
chvarvv |
⊢ ( 𝑖 ∈ 𝐴 ↔ ( 𝑖 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑖 ‘ 𝑥 ) < ( 𝑖 ‘ 𝑦 ) ) ) ) |
136 |
135
|
biimpi |
⊢ ( 𝑖 ∈ 𝐴 → ( 𝑖 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑖 ‘ 𝑥 ) < ( 𝑖 ‘ 𝑦 ) ) ) ) |
137 |
136
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ) → ( 𝑖 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑖 ‘ 𝑥 ) < ( 𝑖 ‘ 𝑦 ) ) ) ) |
138 |
137
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ) → 𝑖 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ) |
139 |
138
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) → 𝑖 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ) |
140 |
139
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) → 𝑖 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ) |
141 |
140
|
frnd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) → ran 𝑖 ⊆ ( 1 ... 𝑁 ) ) |
142 |
125 141
|
sselpwd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) → ran 𝑖 ∈ 𝒫 ( 1 ... 𝑁 ) ) |
143 |
122 124 113 142
|
fvmptd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) → ( 𝐹 ‘ 𝑖 ) = ran 𝑖 ) |
144 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) ∧ 𝑧 = 𝑗 ) → 𝑧 = 𝑗 ) |
145 |
144
|
rneqd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) ∧ 𝑧 = 𝑗 ) → ran 𝑧 = ran 𝑗 ) |
146 |
|
fzfid |
⊢ ( 𝜑 → ( 1 ... 𝑁 ) ∈ Fin ) |
147 |
146
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) → ( 1 ... 𝑁 ) ∈ Fin ) |
148 |
|
eleq1w |
⊢ ( 𝑓 = 𝑗 → ( 𝑓 ∈ 𝐴 ↔ 𝑗 ∈ 𝐴 ) ) |
149 |
|
feq1 |
⊢ ( 𝑓 = 𝑗 → ( 𝑓 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ↔ 𝑗 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ) ) |
150 |
|
fveq1 |
⊢ ( 𝑓 = 𝑗 → ( 𝑓 ‘ 𝑥 ) = ( 𝑗 ‘ 𝑥 ) ) |
151 |
|
fveq1 |
⊢ ( 𝑓 = 𝑗 → ( 𝑓 ‘ 𝑦 ) = ( 𝑗 ‘ 𝑦 ) ) |
152 |
150 151
|
breq12d |
⊢ ( 𝑓 = 𝑗 → ( ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ↔ ( 𝑗 ‘ 𝑥 ) < ( 𝑗 ‘ 𝑦 ) ) ) |
153 |
152
|
imbi2d |
⊢ ( 𝑓 = 𝑗 → ( ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ↔ ( 𝑥 < 𝑦 → ( 𝑗 ‘ 𝑥 ) < ( 𝑗 ‘ 𝑦 ) ) ) ) |
154 |
153
|
2ralbidv |
⊢ ( 𝑓 = 𝑗 → ( ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑗 ‘ 𝑥 ) < ( 𝑗 ‘ 𝑦 ) ) ) ) |
155 |
149 154
|
anbi12d |
⊢ ( 𝑓 = 𝑗 → ( ( 𝑓 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ) ↔ ( 𝑗 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑗 ‘ 𝑥 ) < ( 𝑗 ‘ 𝑦 ) ) ) ) ) |
156 |
148 155
|
bibi12d |
⊢ ( 𝑓 = 𝑗 → ( ( 𝑓 ∈ 𝐴 ↔ ( 𝑓 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑓 ‘ 𝑥 ) < ( 𝑓 ‘ 𝑦 ) ) ) ) ↔ ( 𝑗 ∈ 𝐴 ↔ ( 𝑗 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑗 ‘ 𝑥 ) < ( 𝑗 ‘ 𝑦 ) ) ) ) ) ) |
157 |
156 20
|
chvarvv |
⊢ ( 𝑗 ∈ 𝐴 ↔ ( 𝑗 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑗 ‘ 𝑥 ) < ( 𝑗 ‘ 𝑦 ) ) ) ) |
158 |
157
|
biimpi |
⊢ ( 𝑗 ∈ 𝐴 → ( 𝑗 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑗 ‘ 𝑥 ) < ( 𝑗 ‘ 𝑦 ) ) ) ) |
159 |
158
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ( 𝑗 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ∧ ∀ 𝑥 ∈ ( 1 ... 𝐾 ) ∀ 𝑦 ∈ ( 1 ... 𝐾 ) ( 𝑥 < 𝑦 → ( 𝑗 ‘ 𝑥 ) < ( 𝑗 ‘ 𝑦 ) ) ) ) |
160 |
159
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝑗 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ) |
161 |
160
|
3adant2 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) → 𝑗 : ( 1 ... 𝐾 ) ⟶ ( 1 ... 𝑁 ) ) |
162 |
161
|
frnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) → ran 𝑗 ⊆ ( 1 ... 𝑁 ) ) |
163 |
147 162
|
sselpwd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) → ran 𝑗 ∈ 𝒫 ( 1 ... 𝑁 ) ) |
164 |
163
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) → ran 𝑗 ∈ 𝒫 ( 1 ... 𝑁 ) ) |
165 |
122 145 114 164
|
fvmptd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) → ( 𝐹 ‘ 𝑗 ) = ran 𝑗 ) |
166 |
121 143 165
|
3netr4d |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑖 ≠ 𝑗 ) → ( 𝐹 ‘ 𝑖 ) ≠ ( 𝐹 ‘ 𝑗 ) ) |
167 |
166
|
ex |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) → ( 𝑖 ≠ 𝑗 → ( 𝐹 ‘ 𝑖 ) ≠ ( 𝐹 ‘ 𝑗 ) ) ) |
168 |
167
|
necon4d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑖 ) = ( 𝐹 ‘ 𝑗 ) → 𝑖 = 𝑗 ) ) |
169 |
168
|
3expa |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ) ∧ 𝑗 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑖 ) = ( 𝐹 ‘ 𝑗 ) → 𝑖 = 𝑗 ) ) |
170 |
169
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐴 ) → ∀ 𝑗 ∈ 𝐴 ( ( 𝐹 ‘ 𝑖 ) = ( 𝐹 ‘ 𝑗 ) → 𝑖 = 𝑗 ) ) |
171 |
170
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝐴 ∀ 𝑗 ∈ 𝐴 ( ( 𝐹 ‘ 𝑖 ) = ( 𝐹 ‘ 𝑗 ) → 𝑖 = 𝑗 ) ) |
172 |
108 171
|
jca |
⊢ ( 𝜑 → ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑖 ∈ 𝐴 ∀ 𝑗 ∈ 𝐴 ( ( 𝐹 ‘ 𝑖 ) = ( 𝐹 ‘ 𝑗 ) → 𝑖 = 𝑗 ) ) ) |
173 |
|
dff13 |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑖 ∈ 𝐴 ∀ 𝑗 ∈ 𝐴 ( ( 𝐹 ‘ 𝑖 ) = ( 𝐹 ‘ 𝑗 ) → 𝑖 = 𝑗 ) ) ) |
174 |
172 173
|
sylibr |
⊢ ( 𝜑 → 𝐹 : 𝐴 –1-1→ 𝐵 ) |