Step |
Hyp |
Ref |
Expression |
1 |
|
suppcoss.f |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
2 |
|
suppcoss.g |
⊢ ( 𝜑 → 𝐺 : 𝐵 ⟶ 𝐴 ) |
3 |
|
suppcoss.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
4 |
|
suppcoss.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
5 |
|
suppcoss.1 |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑌 ) = 𝑍 ) |
6 |
|
dffn3 |
⊢ ( 𝐹 Fn 𝐴 ↔ 𝐹 : 𝐴 ⟶ ran 𝐹 ) |
7 |
1 6
|
sylib |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ran 𝐹 ) |
8 |
7 2
|
fcod |
⊢ ( 𝜑 → ( 𝐹 ∘ 𝐺 ) : 𝐵 ⟶ ran 𝐹 ) |
9 |
|
eldif |
⊢ ( 𝑘 ∈ ( 𝐵 ∖ ( 𝐺 supp 𝑌 ) ) ↔ ( 𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ ( 𝐺 supp 𝑌 ) ) ) |
10 |
2
|
ffnd |
⊢ ( 𝜑 → 𝐺 Fn 𝐵 ) |
11 |
|
elsuppfn |
⊢ ( ( 𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑊 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑘 ∈ ( 𝐺 supp 𝑌 ) ↔ ( 𝑘 ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑘 ) ≠ 𝑌 ) ) ) |
12 |
10 3 4 11
|
syl3anc |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 𝐺 supp 𝑌 ) ↔ ( 𝑘 ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑘 ) ≠ 𝑌 ) ) ) |
13 |
12
|
notbid |
⊢ ( 𝜑 → ( ¬ 𝑘 ∈ ( 𝐺 supp 𝑌 ) ↔ ¬ ( 𝑘 ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑘 ) ≠ 𝑌 ) ) ) |
14 |
13
|
anbi2d |
⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ ( 𝐺 supp 𝑌 ) ) ↔ ( 𝑘 ∈ 𝐵 ∧ ¬ ( 𝑘 ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑘 ) ≠ 𝑌 ) ) ) ) |
15 |
|
annotanannot |
⊢ ( ( 𝑘 ∈ 𝐵 ∧ ¬ ( 𝑘 ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑘 ) ≠ 𝑌 ) ) ↔ ( 𝑘 ∈ 𝐵 ∧ ¬ ( 𝐺 ‘ 𝑘 ) ≠ 𝑌 ) ) |
16 |
14 15
|
bitrdi |
⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ ( 𝐺 supp 𝑌 ) ) ↔ ( 𝑘 ∈ 𝐵 ∧ ¬ ( 𝐺 ‘ 𝑘 ) ≠ 𝑌 ) ) ) |
17 |
9 16
|
syl5bb |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 𝐵 ∖ ( 𝐺 supp 𝑌 ) ) ↔ ( 𝑘 ∈ 𝐵 ∧ ¬ ( 𝐺 ‘ 𝑘 ) ≠ 𝑌 ) ) ) |
18 |
|
nne |
⊢ ( ¬ ( 𝐺 ‘ 𝑘 ) ≠ 𝑌 ↔ ( 𝐺 ‘ 𝑘 ) = 𝑌 ) |
19 |
18
|
anbi2i |
⊢ ( ( 𝑘 ∈ 𝐵 ∧ ¬ ( 𝐺 ‘ 𝑘 ) ≠ 𝑌 ) ↔ ( 𝑘 ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑘 ) = 𝑌 ) ) |
20 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑘 ) = 𝑌 ) ) → 𝐺 : 𝐵 ⟶ 𝐴 ) |
21 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑘 ) = 𝑌 ) ) → 𝑘 ∈ 𝐵 ) |
22 |
20 21
|
fvco3d |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑘 ) = 𝑌 ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑘 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑘 ) ) ) |
23 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑘 ) = 𝑌 ) ) → ( 𝐺 ‘ 𝑘 ) = 𝑌 ) |
24 |
23
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑘 ) = 𝑌 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑘 ) ) = ( 𝐹 ‘ 𝑌 ) ) |
25 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑘 ) = 𝑌 ) ) → ( 𝐹 ‘ 𝑌 ) = 𝑍 ) |
26 |
22 24 25
|
3eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑘 ) = 𝑌 ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑘 ) = 𝑍 ) |
27 |
26
|
ex |
⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑘 ) = 𝑌 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑘 ) = 𝑍 ) ) |
28 |
19 27
|
syl5bi |
⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝐵 ∧ ¬ ( 𝐺 ‘ 𝑘 ) ≠ 𝑌 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑘 ) = 𝑍 ) ) |
29 |
17 28
|
sylbid |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 𝐵 ∖ ( 𝐺 supp 𝑌 ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑘 ) = 𝑍 ) ) |
30 |
29
|
imp |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝐵 ∖ ( 𝐺 supp 𝑌 ) ) ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑘 ) = 𝑍 ) |
31 |
8 30
|
suppss |
⊢ ( 𝜑 → ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) ⊆ ( 𝐺 supp 𝑌 ) ) |