| Step |
Hyp |
Ref |
Expression |
| 1 |
|
simplr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) → ¬ 𝑆 ∈ Fin ) |
| 2 |
|
fzfi |
⊢ ( 1 ... 𝑁 ) ∈ Fin |
| 3 |
|
difinf |
⊢ ( ( ¬ 𝑆 ∈ Fin ∧ ( 1 ... 𝑁 ) ∈ Fin ) → ¬ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ∈ Fin ) |
| 4 |
1 2 3
|
sylancl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) → ¬ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ∈ Fin ) |
| 5 |
|
fzfi |
⊢ ( 1 ... 𝐴 ) ∈ Fin |
| 6 |
|
diffi |
⊢ ( ( 1 ... 𝐴 ) ∈ Fin → ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∈ Fin ) |
| 7 |
5 6
|
ax-mp |
⊢ ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∈ Fin |
| 8 |
|
isinffi |
⊢ ( ( ¬ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ∈ Fin ∧ ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∈ Fin ) → ∃ 𝑎 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) |
| 9 |
4 7 8
|
sylancl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) → ∃ 𝑎 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) |
| 10 |
|
f1f1orn |
⊢ ( 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) → 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1-onto→ ran 𝑎 ) |
| 11 |
10
|
adantl |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1-onto→ ran 𝑎 ) |
| 12 |
|
f1oi |
⊢ ( I ↾ ( 1 ... 𝑁 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) |
| 13 |
12
|
a1i |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( I ↾ ( 1 ... 𝑁 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) |
| 14 |
|
disjdifr |
⊢ ( ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∩ ( 1 ... 𝑁 ) ) = ∅ |
| 15 |
14
|
a1i |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∩ ( 1 ... 𝑁 ) ) = ∅ ) |
| 16 |
|
f1f |
⊢ ( 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) → 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ⟶ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) |
| 17 |
16
|
frnd |
⊢ ( 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) → ran 𝑎 ⊆ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) |
| 18 |
17
|
adantl |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ran 𝑎 ⊆ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) |
| 19 |
18
|
ssrind |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( ran 𝑎 ∩ ( 1 ... 𝑁 ) ) ⊆ ( ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ∩ ( 1 ... 𝑁 ) ) ) |
| 20 |
|
disjdifr |
⊢ ( ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ∩ ( 1 ... 𝑁 ) ) = ∅ |
| 21 |
19 20
|
sseqtrdi |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( ran 𝑎 ∩ ( 1 ... 𝑁 ) ) ⊆ ∅ ) |
| 22 |
|
ss0 |
⊢ ( ( ran 𝑎 ∩ ( 1 ... 𝑁 ) ) ⊆ ∅ → ( ran 𝑎 ∩ ( 1 ... 𝑁 ) ) = ∅ ) |
| 23 |
21 22
|
syl |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( ran 𝑎 ∩ ( 1 ... 𝑁 ) ) = ∅ ) |
| 24 |
|
f1oun |
⊢ ( ( ( 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1-onto→ ran 𝑎 ∧ ( I ↾ ( 1 ... 𝑁 ) ) : ( 1 ... 𝑁 ) –1-1-onto→ ( 1 ... 𝑁 ) ) ∧ ( ( ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∩ ( 1 ... 𝑁 ) ) = ∅ ∧ ( ran 𝑎 ∩ ( 1 ... 𝑁 ) ) = ∅ ) ) → ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) –1-1-onto→ ( ran 𝑎 ∪ ( 1 ... 𝑁 ) ) ) |
| 25 |
11 13 15 23 24
|
syl22anc |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) –1-1-onto→ ( ran 𝑎 ∪ ( 1 ... 𝑁 ) ) ) |
| 26 |
|
f1of1 |
⊢ ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) –1-1-onto→ ( ran 𝑎 ∪ ( 1 ... 𝑁 ) ) → ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) –1-1→ ( ran 𝑎 ∪ ( 1 ... 𝑁 ) ) ) |
| 27 |
25 26
|
syl |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) –1-1→ ( ran 𝑎 ∪ ( 1 ... 𝑁 ) ) ) |
| 28 |
|
uncom |
⊢ ( ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) = ( ( 1 ... 𝑁 ) ∪ ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ) |
| 29 |
|
simplrr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
| 30 |
|
fzss2 |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) → ( 1 ... 𝑁 ) ⊆ ( 1 ... 𝐴 ) ) |
| 31 |
29 30
|
syl |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( 1 ... 𝑁 ) ⊆ ( 1 ... 𝐴 ) ) |
| 32 |
|
undif |
⊢ ( ( 1 ... 𝑁 ) ⊆ ( 1 ... 𝐴 ) ↔ ( ( 1 ... 𝑁 ) ∪ ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ) = ( 1 ... 𝐴 ) ) |
| 33 |
31 32
|
sylib |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 1 ... 𝑁 ) ∪ ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ) = ( 1 ... 𝐴 ) ) |
| 34 |
28 33
|
syl5eq |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) = ( 1 ... 𝐴 ) ) |
| 35 |
|
f1eq2 |
⊢ ( ( ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) = ( 1 ... 𝐴 ) → ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) –1-1→ ( ran 𝑎 ∪ ( 1 ... 𝑁 ) ) ↔ ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( 1 ... 𝐴 ) –1-1→ ( ran 𝑎 ∪ ( 1 ... 𝑁 ) ) ) ) |
| 36 |
34 35
|
syl |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∪ ( 1 ... 𝑁 ) ) –1-1→ ( ran 𝑎 ∪ ( 1 ... 𝑁 ) ) ↔ ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( 1 ... 𝐴 ) –1-1→ ( ran 𝑎 ∪ ( 1 ... 𝑁 ) ) ) ) |
| 37 |
27 36
|
mpbid |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( 1 ... 𝐴 ) –1-1→ ( ran 𝑎 ∪ ( 1 ... 𝑁 ) ) ) |
| 38 |
17
|
difss2d |
⊢ ( 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) → ran 𝑎 ⊆ 𝑆 ) |
| 39 |
38
|
adantl |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ran 𝑎 ⊆ 𝑆 ) |
| 40 |
|
simplrl |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( 1 ... 𝑁 ) ⊆ 𝑆 ) |
| 41 |
39 40
|
unssd |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( ran 𝑎 ∪ ( 1 ... 𝑁 ) ) ⊆ 𝑆 ) |
| 42 |
|
f1ss |
⊢ ( ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( 1 ... 𝐴 ) –1-1→ ( ran 𝑎 ∪ ( 1 ... 𝑁 ) ) ∧ ( ran 𝑎 ∪ ( 1 ... 𝑁 ) ) ⊆ 𝑆 ) → ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( 1 ... 𝐴 ) –1-1→ 𝑆 ) |
| 43 |
37 41 42
|
syl2anc |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( 1 ... 𝐴 ) –1-1→ 𝑆 ) |
| 44 |
|
resundir |
⊢ ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ↾ ( 1 ... 𝑁 ) ) = ( ( 𝑎 ↾ ( 1 ... 𝑁 ) ) ∪ ( ( I ↾ ( 1 ... 𝑁 ) ) ↾ ( 1 ... 𝑁 ) ) ) |
| 45 |
|
dmres |
⊢ dom ( 𝑎 ↾ ( 1 ... 𝑁 ) ) = ( ( 1 ... 𝑁 ) ∩ dom 𝑎 ) |
| 46 |
|
incom |
⊢ ( ( 1 ... 𝑁 ) ∩ dom 𝑎 ) = ( dom 𝑎 ∩ ( 1 ... 𝑁 ) ) |
| 47 |
|
f1dm |
⊢ ( 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) → dom 𝑎 = ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ) |
| 48 |
47
|
adantl |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → dom 𝑎 = ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ) |
| 49 |
48
|
ineq1d |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( dom 𝑎 ∩ ( 1 ... 𝑁 ) ) = ( ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) ∩ ( 1 ... 𝑁 ) ) ) |
| 50 |
49 14
|
eqtrdi |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( dom 𝑎 ∩ ( 1 ... 𝑁 ) ) = ∅ ) |
| 51 |
46 50
|
syl5eq |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 1 ... 𝑁 ) ∩ dom 𝑎 ) = ∅ ) |
| 52 |
45 51
|
syl5eq |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → dom ( 𝑎 ↾ ( 1 ... 𝑁 ) ) = ∅ ) |
| 53 |
|
relres |
⊢ Rel ( 𝑎 ↾ ( 1 ... 𝑁 ) ) |
| 54 |
|
reldm0 |
⊢ ( Rel ( 𝑎 ↾ ( 1 ... 𝑁 ) ) → ( ( 𝑎 ↾ ( 1 ... 𝑁 ) ) = ∅ ↔ dom ( 𝑎 ↾ ( 1 ... 𝑁 ) ) = ∅ ) ) |
| 55 |
53 54
|
ax-mp |
⊢ ( ( 𝑎 ↾ ( 1 ... 𝑁 ) ) = ∅ ↔ dom ( 𝑎 ↾ ( 1 ... 𝑁 ) ) = ∅ ) |
| 56 |
52 55
|
sylibr |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( 𝑎 ↾ ( 1 ... 𝑁 ) ) = ∅ ) |
| 57 |
|
residm |
⊢ ( ( I ↾ ( 1 ... 𝑁 ) ) ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) |
| 58 |
57
|
a1i |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( ( I ↾ ( 1 ... 𝑁 ) ) ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) |
| 59 |
56 58
|
uneq12d |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 𝑎 ↾ ( 1 ... 𝑁 ) ) ∪ ( ( I ↾ ( 1 ... 𝑁 ) ) ↾ ( 1 ... 𝑁 ) ) ) = ( ∅ ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ) |
| 60 |
|
uncom |
⊢ ( ∅ ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) = ( ( I ↾ ( 1 ... 𝑁 ) ) ∪ ∅ ) |
| 61 |
|
un0 |
⊢ ( ( I ↾ ( 1 ... 𝑁 ) ) ∪ ∅ ) = ( I ↾ ( 1 ... 𝑁 ) ) |
| 62 |
60 61
|
eqtri |
⊢ ( ∅ ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) = ( I ↾ ( 1 ... 𝑁 ) ) |
| 63 |
59 62
|
eqtrdi |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 𝑎 ↾ ( 1 ... 𝑁 ) ) ∪ ( ( I ↾ ( 1 ... 𝑁 ) ) ↾ ( 1 ... 𝑁 ) ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) |
| 64 |
44 63
|
syl5eq |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) |
| 65 |
|
vex |
⊢ 𝑎 ∈ V |
| 66 |
|
ovex |
⊢ ( 1 ... 𝑁 ) ∈ V |
| 67 |
|
resiexg |
⊢ ( ( 1 ... 𝑁 ) ∈ V → ( I ↾ ( 1 ... 𝑁 ) ) ∈ V ) |
| 68 |
66 67
|
ax-mp |
⊢ ( I ↾ ( 1 ... 𝑁 ) ) ∈ V |
| 69 |
65 68
|
unex |
⊢ ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ∈ V |
| 70 |
|
f1eq1 |
⊢ ( 𝑐 = ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) → ( 𝑐 : ( 1 ... 𝐴 ) –1-1→ 𝑆 ↔ ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( 1 ... 𝐴 ) –1-1→ 𝑆 ) ) |
| 71 |
|
reseq1 |
⊢ ( 𝑐 = ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) → ( 𝑐 ↾ ( 1 ... 𝑁 ) ) = ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ↾ ( 1 ... 𝑁 ) ) ) |
| 72 |
71
|
eqeq1d |
⊢ ( 𝑐 = ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) → ( ( 𝑐 ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ↔ ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) ) |
| 73 |
70 72
|
anbi12d |
⊢ ( 𝑐 = ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) → ( ( 𝑐 : ( 1 ... 𝐴 ) –1-1→ 𝑆 ∧ ( 𝑐 ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) ↔ ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( 1 ... 𝐴 ) –1-1→ 𝑆 ∧ ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) ) ) |
| 74 |
69 73
|
spcev |
⊢ ( ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) : ( 1 ... 𝐴 ) –1-1→ 𝑆 ∧ ( ( 𝑎 ∪ ( I ↾ ( 1 ... 𝑁 ) ) ) ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) → ∃ 𝑐 ( 𝑐 : ( 1 ... 𝐴 ) –1-1→ 𝑆 ∧ ( 𝑐 ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) ) |
| 75 |
43 64 74
|
syl2anc |
⊢ ( ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) ∧ 𝑎 : ( ( 1 ... 𝐴 ) ∖ ( 1 ... 𝑁 ) ) –1-1→ ( 𝑆 ∖ ( 1 ... 𝑁 ) ) ) → ∃ 𝑐 ( 𝑐 : ( 1 ... 𝐴 ) –1-1→ 𝑆 ∧ ( 𝑐 ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) ) |
| 76 |
9 75
|
exlimddv |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin ) ∧ ( ( 1 ... 𝑁 ) ⊆ 𝑆 ∧ 𝐴 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) → ∃ 𝑐 ( 𝑐 : ( 1 ... 𝐴 ) –1-1→ 𝑆 ∧ ( 𝑐 ↾ ( 1 ... 𝑁 ) ) = ( I ↾ ( 1 ... 𝑁 ) ) ) ) |