| Step |
Hyp |
Ref |
Expression |
| 1 |
|
nn0re |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ ) |
| 2 |
1
|
3ad2ant1 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → 𝑁 ∈ ℝ ) |
| 3 |
2
|
recnd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → 𝑁 ∈ ℂ ) |
| 4 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 5 |
|
addcom |
⊢ ( ( 𝑁 ∈ ℂ ∧ 1 ∈ ℂ ) → ( 𝑁 + 1 ) = ( 1 + 𝑁 ) ) |
| 6 |
3 4 5
|
sylancl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( 𝑁 + 1 ) = ( 1 + 𝑁 ) ) |
| 7 |
|
diffi |
⊢ ( 𝐴 ∈ Fin → ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∈ Fin ) |
| 8 |
7
|
3ad2ant2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∈ Fin ) |
| 9 |
|
fzfid |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( 1 ... 𝑁 ) ∈ Fin ) |
| 10 |
|
disjdifr |
⊢ ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∩ ( 1 ... 𝑁 ) ) = ∅ |
| 11 |
10
|
a1i |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∩ ( 1 ... 𝑁 ) ) = ∅ ) |
| 12 |
|
hashun |
⊢ ( ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∈ Fin ∧ ( 1 ... 𝑁 ) ∈ Fin ∧ ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∩ ( 1 ... 𝑁 ) ) = ∅ ) → ( ♯ ‘ ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) ) = ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + ( ♯ ‘ ( 1 ... 𝑁 ) ) ) ) |
| 13 |
8 9 11 12
|
syl3anc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ♯ ‘ ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) ) = ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + ( ♯ ‘ ( 1 ... 𝑁 ) ) ) ) |
| 14 |
|
uncom |
⊢ ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) = ( ( 1 ... 𝑁 ) ∪ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) |
| 15 |
|
simp3 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( 1 ... 𝑁 ) ⊆ 𝐴 ) |
| 16 |
|
undif |
⊢ ( ( 1 ... 𝑁 ) ⊆ 𝐴 ↔ ( ( 1 ... 𝑁 ) ∪ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) = 𝐴 ) |
| 17 |
15 16
|
sylib |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ( 1 ... 𝑁 ) ∪ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) = 𝐴 ) |
| 18 |
14 17
|
syl5eq |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) = 𝐴 ) |
| 19 |
18
|
fveq2d |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ♯ ‘ ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) ) = ( ♯ ‘ 𝐴 ) ) |
| 20 |
|
hashfz1 |
⊢ ( 𝑁 ∈ ℕ0 → ( ♯ ‘ ( 1 ... 𝑁 ) ) = 𝑁 ) |
| 21 |
20
|
3ad2ant1 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ♯ ‘ ( 1 ... 𝑁 ) ) = 𝑁 ) |
| 22 |
21
|
oveq2d |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + ( ♯ ‘ ( 1 ... 𝑁 ) ) ) = ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) |
| 23 |
13 19 22
|
3eqtr3d |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ♯ ‘ 𝐴 ) = ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) |
| 24 |
6 23
|
oveq12d |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) = ( ( 1 + 𝑁 ) ... ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) ) |
| 25 |
24
|
fveq2d |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ♯ ‘ ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ) = ( ♯ ‘ ( ( 1 + 𝑁 ) ... ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) ) ) |
| 26 |
|
1zzd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → 1 ∈ ℤ ) |
| 27 |
|
hashcl |
⊢ ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∈ Fin → ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ∈ ℕ0 ) |
| 28 |
8 27
|
syl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ∈ ℕ0 ) |
| 29 |
28
|
nn0zd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ∈ ℤ ) |
| 30 |
|
nn0z |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ ) |
| 31 |
30
|
3ad2ant1 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → 𝑁 ∈ ℤ ) |
| 32 |
|
fzen |
⊢ ( ( 1 ∈ ℤ ∧ ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 1 ... ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) ≈ ( ( 1 + 𝑁 ) ... ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) ) |
| 33 |
26 29 31 32
|
syl3anc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( 1 ... ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) ≈ ( ( 1 + 𝑁 ) ... ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) ) |
| 34 |
33
|
ensymd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ( 1 + 𝑁 ) ... ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) ≈ ( 1 ... ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) ) |
| 35 |
|
fzfi |
⊢ ( ( 1 + 𝑁 ) ... ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) ∈ Fin |
| 36 |
|
fzfi |
⊢ ( 1 ... ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) ∈ Fin |
| 37 |
|
hashen |
⊢ ( ( ( ( 1 + 𝑁 ) ... ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) ∈ Fin ∧ ( 1 ... ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) ∈ Fin ) → ( ( ♯ ‘ ( ( 1 + 𝑁 ) ... ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) ) = ( ♯ ‘ ( 1 ... ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) ) ↔ ( ( 1 + 𝑁 ) ... ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) ≈ ( 1 ... ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) ) ) |
| 38 |
35 36 37
|
mp2an |
⊢ ( ( ♯ ‘ ( ( 1 + 𝑁 ) ... ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) ) = ( ♯ ‘ ( 1 ... ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) ) ↔ ( ( 1 + 𝑁 ) ... ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) ≈ ( 1 ... ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) ) |
| 39 |
34 38
|
sylibr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ♯ ‘ ( ( 1 + 𝑁 ) ... ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) ) = ( ♯ ‘ ( 1 ... ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) ) ) |
| 40 |
|
hashfz1 |
⊢ ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ∈ ℕ0 → ( ♯ ‘ ( 1 ... ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) ) = ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) |
| 41 |
28 40
|
syl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ♯ ‘ ( 1 ... ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) ) = ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) |
| 42 |
25 39 41
|
3eqtrd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ♯ ‘ ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ) = ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) |
| 43 |
|
fzfi |
⊢ ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ∈ Fin |
| 44 |
|
hashen |
⊢ ( ( ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ∈ Fin ∧ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∈ Fin ) → ( ( ♯ ‘ ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ) = ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ↔ ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ≈ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) |
| 45 |
43 8 44
|
sylancr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ( ♯ ‘ ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ) = ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ↔ ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ≈ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ) |
| 46 |
42 45
|
mpbid |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ≈ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) |
| 47 |
|
bren |
⊢ ( ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ≈ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ↔ ∃ 𝑎 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) |
| 48 |
46 47
|
sylib |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ∃ 𝑎 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) |
| 49 |
|
simpl1 |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → 𝑁 ∈ ℕ0 ) |
| 50 |
49
|
nn0zd |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → 𝑁 ∈ ℤ ) |
| 51 |
|
simpl2 |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → 𝐴 ∈ Fin ) |
| 52 |
|
hashcl |
⊢ ( 𝐴 ∈ Fin → ( ♯ ‘ 𝐴 ) ∈ ℕ0 ) |
| 53 |
51 52
|
syl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ♯ ‘ 𝐴 ) ∈ ℕ0 ) |
| 54 |
53
|
nn0zd |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ♯ ‘ 𝐴 ) ∈ ℤ ) |
| 55 |
|
nn0addge2 |
⊢ ( ( 𝑁 ∈ ℝ ∧ ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) ∈ ℕ0 ) → 𝑁 ≤ ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) |
| 56 |
2 28 55
|
syl2anc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → 𝑁 ≤ ( ( ♯ ‘ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) + 𝑁 ) ) |
| 57 |
56 23
|
breqtrrd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → 𝑁 ≤ ( ♯ ‘ 𝐴 ) ) |
| 58 |
57
|
adantr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → 𝑁 ≤ ( ♯ ‘ 𝐴 ) ) |
| 59 |
|
eluz2 |
⊢ ( ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 𝑁 ) ↔ ( 𝑁 ∈ ℤ ∧ ( ♯ ‘ 𝐴 ) ∈ ℤ ∧ 𝑁 ≤ ( ♯ ‘ 𝐴 ) ) ) |
| 60 |
50 54 58 59
|
syl3anbrc |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 𝑁 ) ) |
| 61 |
|
vex |
⊢ 𝑎 ∈ V |
| 62 |
|
ovex |
⊢ ( 1 ... 𝑁 ) ∈ V |
| 63 |
|
resiexg |
⊢ ( ( 1 ... 𝑁 ) ∈ V → ( I ↾ ( 1 ... 𝑁 ) ) ∈ V ) |
| 64 |
62 63
|
ax-mp |
⊢ ( I ↾ ( 1 ... 𝑁 ) ) ∈ V |
| 65 |
61 64
|
unex |
⊢ ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ∈ V |
| 66 |
65
|
a1i |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ∈ V ) |
| 67 |
|
simpr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) |
| 68 |
|
f1oi |
⊢ ( I ↾ ( 1 ... 𝑁 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) |
| 69 |
68
|
a1i |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( I ↾ ( 1 ... 𝑁 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) |
| 70 |
|
incom |
⊢ ( ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ∩ ( 1 ... 𝑁 ) ) = ( ( 1 ... 𝑁 ) ∩ ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ) |
| 71 |
49
|
nn0red |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → 𝑁 ∈ ℝ ) |
| 72 |
71
|
ltp1d |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → 𝑁 < ( 𝑁 + 1 ) ) |
| 73 |
|
fzdisj |
⊢ ( 𝑁 < ( 𝑁 + 1 ) → ( ( 1 ... 𝑁 ) ∩ ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ) = ∅ ) |
| 74 |
72 73
|
syl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 1 ... 𝑁 ) ∩ ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ) = ∅ ) |
| 75 |
70 74
|
syl5eq |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ∩ ( 1 ... 𝑁 ) ) = ∅ ) |
| 76 |
10
|
a1i |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∩ ( 1 ... 𝑁 ) ) = ∅ ) |
| 77 |
|
f1oun |
⊢ ( ( ( 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∧ ( I ↾ ( 1 ... 𝑁 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) ∧ ( ( ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ∩ ( 1 ... 𝑁 ) ) = ∅ ∧ ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∩ ( 1 ... 𝑁 ) ) = ∅ ) ) → ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ∪ ( 1 ... 𝑁 ) ) –1-1-onto→ ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) ) |
| 78 |
67 69 75 76 77
|
syl22anc |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ∪ ( 1 ... 𝑁 ) ) –1-1-onto→ ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) ) |
| 79 |
|
uncom |
⊢ ( ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ∪ ( 1 ... 𝑁 ) ) = ( ( 1 ... 𝑁 ) ∪ ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ) |
| 80 |
|
fzsplit1nn0 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ0 ∧ 𝑁 ≤ ( ♯ ‘ 𝐴 ) ) → ( 1 ... ( ♯ ‘ 𝐴 ) ) = ( ( 1 ... 𝑁 ) ∪ ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ) ) |
| 81 |
49 53 58 80
|
syl3anc |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( 1 ... ( ♯ ‘ 𝐴 ) ) = ( ( 1 ... 𝑁 ) ∪ ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ) ) |
| 82 |
79 81
|
eqtr4id |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ∪ ( 1 ... 𝑁 ) ) = ( 1 ... ( ♯ ‘ 𝐴 ) ) ) |
| 83 |
|
simpl3 |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( 1 ... 𝑁 ) ⊆ 𝐴 ) |
| 84 |
83 16
|
sylib |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 1 ... 𝑁 ) ∪ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) = 𝐴 ) |
| 85 |
14 84
|
syl5eq |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) = 𝐴 ) |
| 86 |
|
f1oeq23 |
⊢ ( ( ( ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ∪ ( 1 ... 𝑁 ) ) = ( 1 ... ( ♯ ‘ 𝐴 ) ) ∧ ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) = 𝐴 ) → ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ∪ ( 1 ... 𝑁 ) ) –1-1-onto→ ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) ↔ ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) |
| 87 |
82 85 86
|
syl2anc |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ∪ ( 1 ... 𝑁 ) ) –1-1-onto→ ( ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) ↔ ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) |
| 88 |
78 87
|
mpbid |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) |
| 89 |
|
resundir |
⊢ ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ↾ ( 1 ... 𝑁 ) ) = ( ( 𝑎 ↾ ( 1 ... 𝑁 ) ) ∪ ( ( I ↾ ( 1 ... 𝑁 ) ) ↾ ( 1 ... 𝑁 ) ) ) |
| 90 |
|
dmres |
⊢ dom ( 𝑎 ↾ ( 1 ... 𝑁 ) ) = ( ( 1 ... 𝑁 ) ∩ dom 𝑎 ) |
| 91 |
|
f1odm |
⊢ ( 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) → dom 𝑎 = ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ) |
| 92 |
91
|
adantl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → dom 𝑎 = ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ) |
| 93 |
92
|
ineq2d |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 1 ... 𝑁 ) ∩ dom 𝑎 ) = ( ( 1 ... 𝑁 ) ∩ ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) ) ) |
| 94 |
93 74
|
eqtrd |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 1 ... 𝑁 ) ∩ dom 𝑎 ) = ∅ ) |
| 95 |
90 94
|
syl5eq |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → dom ( 𝑎 ↾ ( 1 ... 𝑁 ) ) = ∅ ) |
| 96 |
|
relres |
⊢ Rel ( 𝑎 ↾ ( 1 ... 𝑁 ) ) |
| 97 |
|
reldm0 |
⊢ ( Rel ( 𝑎 ↾ ( 1 ... 𝑁 ) ) → ( ( 𝑎 ↾ ( 1 ... 𝑁 ) ) = ∅ ↔ dom ( 𝑎 ↾ ( 1 ... 𝑁 ) ) = ∅ ) ) |
| 98 |
96 97
|
ax-mp |
⊢ ( ( 𝑎 ↾ ( 1 ... 𝑁 ) ) = ∅ ↔ dom ( 𝑎 ↾ ( 1 ... 𝑁 ) ) = ∅ ) |
| 99 |
95 98
|
sylibr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( 𝑎 ↾ ( 1 ... 𝑁 ) ) = ∅ ) |
| 100 |
|
residm |
⊢ ( ( I ↾ ( 1 ... 𝑁 ) ) ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) |
| 101 |
100
|
a1i |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ( I ↾ ( 1 ... 𝑁 ) ) ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) |
| 102 |
99 101
|
uneq12d |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 𝑎 ↾ ( 1 ... 𝑁 ) ) ∪ ( ( I ↾ ( 1 ... 𝑁 ) ) ↾ ( 1 ... 𝑁 ) ) ) = ( ∅ ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ) |
| 103 |
|
uncom |
⊢ ( ∅ ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) = ( ( I ↾ ( 1 ... 𝑁 ) ) ∪ ∅ ) |
| 104 |
|
un0 |
⊢ ( ( I ↾ ( 1 ... 𝑁 ) ) ∪ ∅ ) = ( I ↾ ( 1 ... 𝑁 ) ) |
| 105 |
103 104
|
eqtri |
⊢ ( ∅ ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) = ( I ↾ ( 1 ... 𝑁 ) ) |
| 106 |
102 105
|
eqtrdi |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 𝑎 ↾ ( 1 ... 𝑁 ) ) ∪ ( ( I ↾ ( 1 ... 𝑁 ) ) ↾ ( 1 ... 𝑁 ) ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) |
| 107 |
89 106
|
syl5eq |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) |
| 108 |
|
oveq2 |
⊢ ( 𝑑 = ( ♯ ‘ 𝐴 ) → ( 1 ... 𝑑 ) = ( 1 ... ( ♯ ‘ 𝐴 ) ) ) |
| 109 |
108
|
f1oeq2d |
⊢ ( 𝑑 = ( ♯ ‘ 𝐴 ) → ( 𝑒 : ( 1 ... 𝑑 ) –1-1-onto→ 𝐴 ↔ 𝑒 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) |
| 110 |
109
|
anbi1d |
⊢ ( 𝑑 = ( ♯ ‘ 𝐴 ) → ( ( 𝑒 : ( 1 ... 𝑑 ) –1-1-onto→ 𝐴 ∧ ( 𝑒 ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) ↔ ( 𝑒 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ∧ ( 𝑒 ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) ) ) |
| 111 |
|
f1oeq1 |
⊢ ( 𝑒 = ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) → ( 𝑒 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ↔ ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) |
| 112 |
|
reseq1 |
⊢ ( 𝑒 = ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) → ( 𝑒 ↾ ( 1 ... 𝑁 ) ) = ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ↾ ( 1 ... 𝑁 ) ) ) |
| 113 |
112
|
eqeq1d |
⊢ ( 𝑒 = ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) → ( ( 𝑒 ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ↔ ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) ) |
| 114 |
111 113
|
anbi12d |
⊢ ( 𝑒 = ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) → ( ( 𝑒 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ∧ ( 𝑒 ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) ↔ ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ∧ ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) ) ) |
| 115 |
110 114
|
rspc2ev |
⊢ ( ( ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 𝑁 ) ∧ ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ∈ V ∧ ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ∧ ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) ) → ∃ 𝑑 ∈ ( ℤ≥ ‘ 𝑁 ) ∃ 𝑒 ∈ V ( 𝑒 : ( 1 ... 𝑑 ) –1-1-onto→ 𝐴 ∧ ( 𝑒 ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) ) |
| 116 |
60 66 88 107 115
|
syl112anc |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) ∧ 𝑎 : ( ( 𝑁 + 1 ) ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ ( 𝐴 ∖ ( 1 ... 𝑁 ) ) ) → ∃ 𝑑 ∈ ( ℤ≥ ‘ 𝑁 ) ∃ 𝑒 ∈ V ( 𝑒 : ( 1 ... 𝑑 ) –1-1-onto→ 𝐴 ∧ ( 𝑒 ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) ) |
| 117 |
48 116
|
exlimddv |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ ( 1 ... 𝑁 ) ⊆ 𝐴 ) → ∃ 𝑑 ∈ ( ℤ≥ ‘ 𝑁 ) ∃ 𝑒 ∈ V ( 𝑒 : ( 1 ... 𝑑 ) –1-1-onto→ 𝐴 ∧ ( 𝑒 ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) ) |