Step |
Hyp |
Ref |
Expression |
1 |
|
fcores.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
2 |
|
fcores.e |
⊢ 𝐸 = ( ran 𝐹 ∩ 𝐶 ) |
3 |
|
fcores.p |
⊢ 𝑃 = ( ◡ 𝐹 “ 𝐶 ) |
4 |
|
fcores.x |
⊢ 𝑋 = ( 𝐹 ↾ 𝑃 ) |
5 |
|
fcores.g |
⊢ ( 𝜑 → 𝐺 : 𝐶 ⟶ 𝐷 ) |
6 |
|
fcores.y |
⊢ 𝑌 = ( 𝐺 ↾ 𝐸 ) |
7 |
5
|
ffnd |
⊢ ( 𝜑 → 𝐺 Fn 𝐶 ) |
8 |
2
|
a1i |
⊢ ( 𝜑 → 𝐸 = ( ran 𝐹 ∩ 𝐶 ) ) |
9 |
|
inss2 |
⊢ ( ran 𝐹 ∩ 𝐶 ) ⊆ 𝐶 |
10 |
8 9
|
eqsstrdi |
⊢ ( 𝜑 → 𝐸 ⊆ 𝐶 ) |
11 |
7 10
|
fnssresd |
⊢ ( 𝜑 → ( 𝐺 ↾ 𝐸 ) Fn 𝐸 ) |
12 |
6
|
fneq1i |
⊢ ( 𝑌 Fn 𝐸 ↔ ( 𝐺 ↾ 𝐸 ) Fn 𝐸 ) |
13 |
11 12
|
sylibr |
⊢ ( 𝜑 → 𝑌 Fn 𝐸 ) |
14 |
1 2 3 4
|
fcoreslem3 |
⊢ ( 𝜑 → 𝑋 : 𝑃 –onto→ 𝐸 ) |
15 |
|
fofn |
⊢ ( 𝑋 : 𝑃 –onto→ 𝐸 → 𝑋 Fn 𝑃 ) |
16 |
14 15
|
syl |
⊢ ( 𝜑 → 𝑋 Fn 𝑃 ) |
17 |
1 2 3 4
|
fcoreslem2 |
⊢ ( 𝜑 → ran 𝑋 = 𝐸 ) |
18 |
|
eqimss |
⊢ ( ran 𝑋 = 𝐸 → ran 𝑋 ⊆ 𝐸 ) |
19 |
17 18
|
syl |
⊢ ( 𝜑 → ran 𝑋 ⊆ 𝐸 ) |
20 |
|
fnco |
⊢ ( ( 𝑌 Fn 𝐸 ∧ 𝑋 Fn 𝑃 ∧ ran 𝑋 ⊆ 𝐸 ) → ( 𝑌 ∘ 𝑋 ) Fn 𝑃 ) |
21 |
13 16 19 20
|
syl3anc |
⊢ ( 𝜑 → ( 𝑌 ∘ 𝑋 ) Fn 𝑃 ) |