Step |
Hyp |
Ref |
Expression |
1 |
|
suppun2.1 |
⊢ ( 𝜑 → 𝐹 ∈ 𝑉 ) |
2 |
|
suppun2.2 |
⊢ ( 𝜑 → 𝐺 ∈ 𝑊 ) |
3 |
|
suppun2.3 |
⊢ ( 𝜑 → 𝑍 ∈ 𝑋 ) |
4 |
|
cnvun |
⊢ ◡ ( 𝐹 ∪ 𝐺 ) = ( ◡ 𝐹 ∪ ◡ 𝐺 ) |
5 |
4
|
imaeq1i |
⊢ ( ◡ ( 𝐹 ∪ 𝐺 ) “ ( V ∖ { 𝑍 } ) ) = ( ( ◡ 𝐹 ∪ ◡ 𝐺 ) “ ( V ∖ { 𝑍 } ) ) |
6 |
|
imaundir |
⊢ ( ( ◡ 𝐹 ∪ ◡ 𝐺 ) “ ( V ∖ { 𝑍 } ) ) = ( ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ∪ ( ◡ 𝐺 “ ( V ∖ { 𝑍 } ) ) ) |
7 |
5 6
|
eqtri |
⊢ ( ◡ ( 𝐹 ∪ 𝐺 ) “ ( V ∖ { 𝑍 } ) ) = ( ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ∪ ( ◡ 𝐺 “ ( V ∖ { 𝑍 } ) ) ) |
8 |
1 2
|
unexd |
⊢ ( 𝜑 → ( 𝐹 ∪ 𝐺 ) ∈ V ) |
9 |
|
suppimacnv |
⊢ ( ( ( 𝐹 ∪ 𝐺 ) ∈ V ∧ 𝑍 ∈ 𝑋 ) → ( ( 𝐹 ∪ 𝐺 ) supp 𝑍 ) = ( ◡ ( 𝐹 ∪ 𝐺 ) “ ( V ∖ { 𝑍 } ) ) ) |
10 |
8 3 9
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐹 ∪ 𝐺 ) supp 𝑍 ) = ( ◡ ( 𝐹 ∪ 𝐺 ) “ ( V ∖ { 𝑍 } ) ) ) |
11 |
|
suppimacnv |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑋 ) → ( 𝐹 supp 𝑍 ) = ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
12 |
1 3 11
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 supp 𝑍 ) = ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
13 |
|
suppimacnv |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑍 ∈ 𝑋 ) → ( 𝐺 supp 𝑍 ) = ( ◡ 𝐺 “ ( V ∖ { 𝑍 } ) ) ) |
14 |
2 3 13
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 supp 𝑍 ) = ( ◡ 𝐺 “ ( V ∖ { 𝑍 } ) ) ) |
15 |
12 14
|
uneq12d |
⊢ ( 𝜑 → ( ( 𝐹 supp 𝑍 ) ∪ ( 𝐺 supp 𝑍 ) ) = ( ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ∪ ( ◡ 𝐺 “ ( V ∖ { 𝑍 } ) ) ) ) |
16 |
7 10 15
|
3eqtr4a |
⊢ ( 𝜑 → ( ( 𝐹 ∪ 𝐺 ) supp 𝑍 ) = ( ( 𝐹 supp 𝑍 ) ∪ ( 𝐺 supp 𝑍 ) ) ) |