Step |
Hyp |
Ref |
Expression |
1 |
|
fargshift.g |
⊢ 𝐺 = ( 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ↦ ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) |
2 |
|
fof |
⊢ ( 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 → 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) |
3 |
1
|
fargshiftf |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) → 𝐺 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ) |
4 |
2 3
|
sylan2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ) → 𝐺 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ) |
5 |
1
|
rnmpt |
⊢ ran 𝐺 = { 𝑦 ∣ ∃ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) } |
6 |
|
fofn |
⊢ ( 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 → 𝐹 Fn ( 1 ... 𝑁 ) ) |
7 |
|
fnrnfv |
⊢ ( 𝐹 Fn ( 1 ... 𝑁 ) → ran 𝐹 = { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } ) |
8 |
6 7
|
syl |
⊢ ( 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 → ran 𝐹 = { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } ) |
9 |
8
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ) → ran 𝐹 = { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } ) |
10 |
|
df-fo |
⊢ ( 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ↔ ( 𝐹 Fn ( 1 ... 𝑁 ) ∧ ran 𝐹 = dom 𝐸 ) ) |
11 |
10
|
biimpi |
⊢ ( 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 → ( 𝐹 Fn ( 1 ... 𝑁 ) ∧ ran 𝐹 = dom 𝐸 ) ) |
12 |
11
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ) → ( 𝐹 Fn ( 1 ... 𝑁 ) ∧ ran 𝐹 = dom 𝐸 ) ) |
13 |
|
eqeq1 |
⊢ ( ran 𝐹 = dom 𝐸 → ( ran 𝐹 = { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } ↔ dom 𝐸 = { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } ) ) |
14 |
|
eqcom |
⊢ ( dom 𝐸 = { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } ↔ { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } = dom 𝐸 ) |
15 |
13 14
|
bitrdi |
⊢ ( ran 𝐹 = dom 𝐸 → ( ran 𝐹 = { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } ↔ { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } = dom 𝐸 ) ) |
16 |
|
ffn |
⊢ ( 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 → 𝐹 Fn ( 1 ... 𝑁 ) ) |
17 |
|
fseq1hash |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 Fn ( 1 ... 𝑁 ) ) → ( ♯ ‘ 𝐹 ) = 𝑁 ) |
18 |
16 17
|
sylan2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) ⟶ dom 𝐸 ) → ( ♯ ‘ 𝐹 ) = 𝑁 ) |
19 |
2 18
|
sylan2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ) → ( ♯ ‘ 𝐹 ) = 𝑁 ) |
20 |
|
fz0add1fz1 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 0 ..^ 𝑁 ) ) → ( 𝑥 + 1 ) ∈ ( 1 ... 𝑁 ) ) |
21 |
|
nn0z |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ ) |
22 |
|
fzval3 |
⊢ ( 𝑁 ∈ ℤ → ( 1 ... 𝑁 ) = ( 1 ..^ ( 𝑁 + 1 ) ) ) |
23 |
21 22
|
syl |
⊢ ( 𝑁 ∈ ℕ0 → ( 1 ... 𝑁 ) = ( 1 ..^ ( 𝑁 + 1 ) ) ) |
24 |
|
nn0cn |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ ) |
25 |
|
1cnd |
⊢ ( 𝑁 ∈ ℕ0 → 1 ∈ ℂ ) |
26 |
24 25
|
addcomd |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) = ( 1 + 𝑁 ) ) |
27 |
26
|
oveq2d |
⊢ ( 𝑁 ∈ ℕ0 → ( 1 ..^ ( 𝑁 + 1 ) ) = ( 1 ..^ ( 1 + 𝑁 ) ) ) |
28 |
23 27
|
eqtrd |
⊢ ( 𝑁 ∈ ℕ0 → ( 1 ... 𝑁 ) = ( 1 ..^ ( 1 + 𝑁 ) ) ) |
29 |
28
|
eleq2d |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑧 ∈ ( 1 ... 𝑁 ) ↔ 𝑧 ∈ ( 1 ..^ ( 1 + 𝑁 ) ) ) ) |
30 |
29
|
biimpa |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑧 ∈ ( 1 ... 𝑁 ) ) → 𝑧 ∈ ( 1 ..^ ( 1 + 𝑁 ) ) ) |
31 |
21
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑧 ∈ ( 1 ... 𝑁 ) ) → 𝑁 ∈ ℤ ) |
32 |
|
fzosubel3 |
⊢ ( ( 𝑧 ∈ ( 1 ..^ ( 1 + 𝑁 ) ) ∧ 𝑁 ∈ ℤ ) → ( 𝑧 − 1 ) ∈ ( 0 ..^ 𝑁 ) ) |
33 |
30 31 32
|
syl2anc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑧 ∈ ( 1 ... 𝑁 ) ) → ( 𝑧 − 1 ) ∈ ( 0 ..^ 𝑁 ) ) |
34 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝑧 − 1 ) → ( 𝑥 + 1 ) = ( ( 𝑧 − 1 ) + 1 ) ) |
35 |
34
|
eqeq2d |
⊢ ( 𝑥 = ( 𝑧 − 1 ) → ( 𝑧 = ( 𝑥 + 1 ) ↔ 𝑧 = ( ( 𝑧 − 1 ) + 1 ) ) ) |
36 |
35
|
adantl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑧 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑥 = ( 𝑧 − 1 ) ) → ( 𝑧 = ( 𝑥 + 1 ) ↔ 𝑧 = ( ( 𝑧 − 1 ) + 1 ) ) ) |
37 |
|
elfzelz |
⊢ ( 𝑧 ∈ ( 1 ... 𝑁 ) → 𝑧 ∈ ℤ ) |
38 |
37
|
zcnd |
⊢ ( 𝑧 ∈ ( 1 ... 𝑁 ) → 𝑧 ∈ ℂ ) |
39 |
38
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑧 ∈ ( 1 ... 𝑁 ) ) → 𝑧 ∈ ℂ ) |
40 |
|
1cnd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑧 ∈ ( 1 ... 𝑁 ) ) → 1 ∈ ℂ ) |
41 |
39 40
|
npcand |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑧 ∈ ( 1 ... 𝑁 ) ) → ( ( 𝑧 − 1 ) + 1 ) = 𝑧 ) |
42 |
41
|
eqcomd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑧 ∈ ( 1 ... 𝑁 ) ) → 𝑧 = ( ( 𝑧 − 1 ) + 1 ) ) |
43 |
33 36 42
|
rspcedvd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑧 ∈ ( 1 ... 𝑁 ) ) → ∃ 𝑥 ∈ ( 0 ..^ 𝑁 ) 𝑧 = ( 𝑥 + 1 ) ) |
44 |
|
fveq2 |
⊢ ( 𝑧 = ( 𝑥 + 1 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) |
45 |
44
|
eqeq2d |
⊢ ( 𝑧 = ( 𝑥 + 1 ) → ( 𝑦 = ( 𝐹 ‘ 𝑧 ) ↔ 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ) |
46 |
45
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑧 = ( 𝑥 + 1 ) ) → ( 𝑦 = ( 𝐹 ‘ 𝑧 ) ↔ 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ) |
47 |
20 43 46
|
rexxfrd |
⊢ ( 𝑁 ∈ ℕ0 → ( ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) ↔ ∃ 𝑥 ∈ ( 0 ..^ 𝑁 ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ) |
48 |
47
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( ♯ ‘ 𝐹 ) = 𝑁 ) → ( ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) ↔ ∃ 𝑥 ∈ ( 0 ..^ 𝑁 ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ) |
49 |
|
oveq2 |
⊢ ( ( ♯ ‘ 𝐹 ) = 𝑁 → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 𝑁 ) ) |
50 |
49
|
rexeqdv |
⊢ ( ( ♯ ‘ 𝐹 ) = 𝑁 → ( ∃ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) ↔ ∃ 𝑥 ∈ ( 0 ..^ 𝑁 ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ) |
51 |
50
|
bibi2d |
⊢ ( ( ♯ ‘ 𝐹 ) = 𝑁 → ( ( ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) ↔ ∃ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ↔ ( ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) ↔ ∃ 𝑥 ∈ ( 0 ..^ 𝑁 ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ) ) |
52 |
51
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( ♯ ‘ 𝐹 ) = 𝑁 ) → ( ( ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) ↔ ∃ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ↔ ( ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) ↔ ∃ 𝑥 ∈ ( 0 ..^ 𝑁 ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ) ) |
53 |
48 52
|
mpbird |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( ♯ ‘ 𝐹 ) = 𝑁 ) → ( ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) ↔ ∃ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ) |
54 |
19 53
|
syldan |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ) → ( ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) ↔ ∃ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ) |
55 |
54
|
abbidv |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ) → { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } = { 𝑦 ∣ ∃ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) } ) |
56 |
55
|
eqeq1d |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ) → ( { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } = dom 𝐸 ↔ { 𝑦 ∣ ∃ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) } = dom 𝐸 ) ) |
57 |
56
|
biimpcd |
⊢ ( { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } = dom 𝐸 → ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ) → { 𝑦 ∣ ∃ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) } = dom 𝐸 ) ) |
58 |
15 57
|
syl6bi |
⊢ ( ran 𝐹 = dom 𝐸 → ( ran 𝐹 = { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } → ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ) → { 𝑦 ∣ ∃ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) } = dom 𝐸 ) ) ) |
59 |
58
|
com23 |
⊢ ( ran 𝐹 = dom 𝐸 → ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ) → ( ran 𝐹 = { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } → { 𝑦 ∣ ∃ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) } = dom 𝐸 ) ) ) |
60 |
59
|
adantl |
⊢ ( ( 𝐹 Fn ( 1 ... 𝑁 ) ∧ ran 𝐹 = dom 𝐸 ) → ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ) → ( ran 𝐹 = { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } → { 𝑦 ∣ ∃ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) } = dom 𝐸 ) ) ) |
61 |
12 60
|
mpcom |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ) → ( ran 𝐹 = { 𝑦 ∣ ∃ 𝑧 ∈ ( 1 ... 𝑁 ) 𝑦 = ( 𝐹 ‘ 𝑧 ) } → { 𝑦 ∣ ∃ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) } = dom 𝐸 ) ) |
62 |
9 61
|
mpd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ) → { 𝑦 ∣ ∃ 𝑥 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 𝑦 = ( 𝐹 ‘ ( 𝑥 + 1 ) ) } = dom 𝐸 ) |
63 |
5 62
|
syl5eq |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ) → ran 𝐺 = dom 𝐸 ) |
64 |
|
dffo2 |
⊢ ( 𝐺 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –onto→ dom 𝐸 ↔ ( 𝐺 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐸 ∧ ran 𝐺 = dom 𝐸 ) ) |
65 |
4 63 64
|
sylanbrc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐹 : ( 1 ... 𝑁 ) –onto→ dom 𝐸 ) → 𝐺 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –onto→ dom 𝐸 ) |