Step |
Hyp |
Ref |
Expression |
1 |
|
ffun |
⊢ ( 𝐹 : 𝐼 ⟶ 𝑆 → Fun 𝐹 ) |
2 |
1
|
adantl |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐹 : 𝐼 ⟶ 𝑆 ) → Fun 𝐹 ) |
3 |
|
fex |
⊢ ( ( 𝐹 : 𝐼 ⟶ 𝑆 ∧ 𝐼 ∈ 𝑉 ) → 𝐹 ∈ V ) |
4 |
3
|
expcom |
⊢ ( 𝐼 ∈ 𝑉 → ( 𝐹 : 𝐼 ⟶ 𝑆 → 𝐹 ∈ V ) ) |
5 |
4
|
adantr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( 𝐹 : 𝐼 ⟶ 𝑆 → 𝐹 ∈ V ) ) |
6 |
5
|
imp |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐹 : 𝐼 ⟶ 𝑆 ) → 𝐹 ∈ V ) |
7 |
|
simplr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐹 : 𝐼 ⟶ 𝑆 ) → 𝑍 ∈ 𝑊 ) |
8 |
|
funisfsupp |
⊢ ( ( Fun 𝐹 ∧ 𝐹 ∈ V ∧ 𝑍 ∈ 𝑊 ) → ( 𝐹 finSupp 𝑍 ↔ ( 𝐹 supp 𝑍 ) ∈ Fin ) ) |
9 |
2 6 7 8
|
syl3anc |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐹 : 𝐼 ⟶ 𝑆 ) → ( 𝐹 finSupp 𝑍 ↔ ( 𝐹 supp 𝑍 ) ∈ Fin ) ) |
10 |
|
frnsuppeq |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( 𝐹 : 𝐼 ⟶ 𝑆 → ( 𝐹 supp 𝑍 ) = ( ◡ 𝐹 “ ( 𝑆 ∖ { 𝑍 } ) ) ) ) |
11 |
10
|
imp |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐹 : 𝐼 ⟶ 𝑆 ) → ( 𝐹 supp 𝑍 ) = ( ◡ 𝐹 “ ( 𝑆 ∖ { 𝑍 } ) ) ) |
12 |
11
|
eleq1d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐹 : 𝐼 ⟶ 𝑆 ) → ( ( 𝐹 supp 𝑍 ) ∈ Fin ↔ ( ◡ 𝐹 “ ( 𝑆 ∖ { 𝑍 } ) ) ∈ Fin ) ) |
13 |
9 12
|
bitrd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐹 : 𝐼 ⟶ 𝑆 ) → ( 𝐹 finSupp 𝑍 ↔ ( ◡ 𝐹 “ ( 𝑆 ∖ { 𝑍 } ) ) ∈ Fin ) ) |
14 |
13
|
ex |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( 𝐹 : 𝐼 ⟶ 𝑆 → ( 𝐹 finSupp 𝑍 ↔ ( ◡ 𝐹 “ ( 𝑆 ∖ { 𝑍 } ) ) ∈ Fin ) ) ) |