Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) ∧ 𝐹 finSupp 𝑍 ) → 𝐹 finSupp 𝑍 ) |
2 |
1
|
fsuppimpd |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) ∧ 𝐹 finSupp 𝑍 ) → ( 𝐹 supp 𝑍 ) ∈ Fin ) |
3 |
2
|
ex |
⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ( 𝐹 finSupp 𝑍 → ( 𝐹 supp 𝑍 ) ∈ Fin ) ) |
4 |
|
elmapfn |
⊢ ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) → 𝐹 Fn ℕ0 ) |
5 |
4
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → 𝐹 Fn ℕ0 ) |
6 |
|
nn0ex |
⊢ ℕ0 ∈ V |
7 |
6
|
a1i |
⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ℕ0 ∈ V ) |
8 |
|
simpr |
⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → 𝑍 ∈ 𝑉 ) |
9 |
|
suppvalfn |
⊢ ( ( 𝐹 Fn ℕ0 ∧ ℕ0 ∈ V ∧ 𝑍 ∈ 𝑉 ) → ( 𝐹 supp 𝑍 ) = { 𝑥 ∈ ℕ0 ∣ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 } ) |
10 |
5 7 8 9
|
syl3anc |
⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ( 𝐹 supp 𝑍 ) = { 𝑥 ∈ ℕ0 ∣ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 } ) |
11 |
10
|
eleq1d |
⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ( ( 𝐹 supp 𝑍 ) ∈ Fin ↔ { 𝑥 ∈ ℕ0 ∣ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 } ∈ Fin ) ) |
12 |
|
rabssnn0fi |
⊢ ( { 𝑥 ∈ ℕ0 ∣ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 } ∈ Fin ↔ ∃ 𝑚 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ¬ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ) ) |
13 |
|
nne |
⊢ ( ¬ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ↔ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) |
14 |
13
|
imbi2i |
⊢ ( ( 𝑚 < 𝑥 → ¬ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ) ↔ ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) |
15 |
14
|
ralbii |
⊢ ( ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ¬ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ) ↔ ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) |
16 |
15
|
rexbii |
⊢ ( ∃ 𝑚 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ¬ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ) ↔ ∃ 𝑚 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) |
17 |
12 16
|
sylbb |
⊢ ( { 𝑥 ∈ ℕ0 ∣ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 } ∈ Fin → ∃ 𝑚 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) |
18 |
11 17
|
syl6bi |
⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ( ( 𝐹 supp 𝑍 ) ∈ Fin → ∃ 𝑚 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ) |
19 |
3 18
|
syld |
⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ( 𝐹 finSupp 𝑍 → ∃ 𝑚 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ) |