Step |
Hyp |
Ref |
Expression |
1 |
|
suppimacnv |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( 𝐹 supp 𝑍 ) = ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
2 |
|
ffun |
⊢ ( 𝐹 : 𝐼 ⟶ 𝑆 → Fun 𝐹 ) |
3 |
|
inpreima |
⊢ ( Fun 𝐹 → ( ◡ 𝐹 “ ( 𝑆 ∩ ( V ∖ { 𝑍 } ) ) ) = ( ( ◡ 𝐹 “ 𝑆 ) ∩ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ) |
4 |
2 3
|
syl |
⊢ ( 𝐹 : 𝐼 ⟶ 𝑆 → ( ◡ 𝐹 “ ( 𝑆 ∩ ( V ∖ { 𝑍 } ) ) ) = ( ( ◡ 𝐹 “ 𝑆 ) ∩ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ) |
5 |
|
cnvimass |
⊢ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ dom 𝐹 |
6 |
|
fdm |
⊢ ( 𝐹 : 𝐼 ⟶ 𝑆 → dom 𝐹 = 𝐼 ) |
7 |
|
fimacnv |
⊢ ( 𝐹 : 𝐼 ⟶ 𝑆 → ( ◡ 𝐹 “ 𝑆 ) = 𝐼 ) |
8 |
6 7
|
eqtr4d |
⊢ ( 𝐹 : 𝐼 ⟶ 𝑆 → dom 𝐹 = ( ◡ 𝐹 “ 𝑆 ) ) |
9 |
5 8
|
sseqtrid |
⊢ ( 𝐹 : 𝐼 ⟶ 𝑆 → ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ ( ◡ 𝐹 “ 𝑆 ) ) |
10 |
|
sseqin2 |
⊢ ( ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ ( ◡ 𝐹 “ 𝑆 ) ↔ ( ( ◡ 𝐹 “ 𝑆 ) ∩ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) = ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
11 |
9 10
|
sylib |
⊢ ( 𝐹 : 𝐼 ⟶ 𝑆 → ( ( ◡ 𝐹 “ 𝑆 ) ∩ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) = ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
12 |
4 11
|
eqtrd |
⊢ ( 𝐹 : 𝐼 ⟶ 𝑆 → ( ◡ 𝐹 “ ( 𝑆 ∩ ( V ∖ { 𝑍 } ) ) ) = ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
13 |
|
invdif |
⊢ ( 𝑆 ∩ ( V ∖ { 𝑍 } ) ) = ( 𝑆 ∖ { 𝑍 } ) |
14 |
13
|
imaeq2i |
⊢ ( ◡ 𝐹 “ ( 𝑆 ∩ ( V ∖ { 𝑍 } ) ) ) = ( ◡ 𝐹 “ ( 𝑆 ∖ { 𝑍 } ) ) |
15 |
12 14
|
eqtr3di |
⊢ ( 𝐹 : 𝐼 ⟶ 𝑆 → ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) = ( ◡ 𝐹 “ ( 𝑆 ∖ { 𝑍 } ) ) ) |
16 |
1 15
|
sylan9eq |
⊢ ( ( ( 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) ∧ 𝐹 : 𝐼 ⟶ 𝑆 ) → ( 𝐹 supp 𝑍 ) = ( ◡ 𝐹 “ ( 𝑆 ∖ { 𝑍 } ) ) ) |
17 |
16
|
ex |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( 𝐹 : 𝐼 ⟶ 𝑆 → ( 𝐹 supp 𝑍 ) = ( ◡ 𝐹 “ ( 𝑆 ∖ { 𝑍 } ) ) ) ) |