Step |
Hyp |
Ref |
Expression |
1 |
|
funcid.b |
⊢ 𝐵 = ( Base ‘ 𝐷 ) |
2 |
|
funcid.1 |
⊢ 1 = ( Id ‘ 𝐷 ) |
3 |
|
funcid.i |
⊢ 𝐼 = ( Id ‘ 𝐸 ) |
4 |
|
funcid.f |
⊢ ( 𝜑 → 𝐹 ( 𝐷 Func 𝐸 ) 𝐺 ) |
5 |
|
funcid.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
id |
⊢ ( 𝑥 = 𝑋 → 𝑥 = 𝑋 ) |
7 |
6 6
|
oveq12d |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 𝐺 𝑥 ) = ( 𝑋 𝐺 𝑋 ) ) |
8 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 1 ‘ 𝑥 ) = ( 1 ‘ 𝑋 ) ) |
9 |
7 8
|
fveq12d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 𝐺 𝑥 ) ‘ ( 1 ‘ 𝑥 ) ) = ( ( 𝑋 𝐺 𝑋 ) ‘ ( 1 ‘ 𝑋 ) ) ) |
10 |
|
2fveq3 |
⊢ ( 𝑥 = 𝑋 → ( 𝐼 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 𝑋 ) ) ) |
11 |
9 10
|
eqeq12d |
⊢ ( 𝑥 = 𝑋 → ( ( ( 𝑥 𝐺 𝑥 ) ‘ ( 1 ‘ 𝑥 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( 𝑋 𝐺 𝑋 ) ‘ ( 1 ‘ 𝑋 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 𝑋 ) ) ) ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝐸 ) = ( Base ‘ 𝐸 ) |
13 |
|
eqid |
⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 ) |
14 |
|
eqid |
⊢ ( Hom ‘ 𝐸 ) = ( Hom ‘ 𝐸 ) |
15 |
|
eqid |
⊢ ( comp ‘ 𝐷 ) = ( comp ‘ 𝐷 ) |
16 |
|
eqid |
⊢ ( comp ‘ 𝐸 ) = ( comp ‘ 𝐸 ) |
17 |
|
df-br |
⊢ ( 𝐹 ( 𝐷 Func 𝐸 ) 𝐺 ↔ 〈 𝐹 , 𝐺 〉 ∈ ( 𝐷 Func 𝐸 ) ) |
18 |
4 17
|
sylib |
⊢ ( 𝜑 → 〈 𝐹 , 𝐺 〉 ∈ ( 𝐷 Func 𝐸 ) ) |
19 |
|
funcrcl |
⊢ ( 〈 𝐹 , 𝐺 〉 ∈ ( 𝐷 Func 𝐸 ) → ( 𝐷 ∈ Cat ∧ 𝐸 ∈ Cat ) ) |
20 |
18 19
|
syl |
⊢ ( 𝜑 → ( 𝐷 ∈ Cat ∧ 𝐸 ∈ Cat ) ) |
21 |
20
|
simpld |
⊢ ( 𝜑 → 𝐷 ∈ Cat ) |
22 |
20
|
simprd |
⊢ ( 𝜑 → 𝐸 ∈ Cat ) |
23 |
1 12 13 14 2 3 15 16 21 22
|
isfunc |
⊢ ( 𝜑 → ( 𝐹 ( 𝐷 Func 𝐸 ) 𝐺 ↔ ( 𝐹 : 𝐵 ⟶ ( Base ‘ 𝐸 ) ∧ 𝐺 ∈ X 𝑧 ∈ ( 𝐵 × 𝐵 ) ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐸 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( ( Hom ‘ 𝐷 ) ‘ 𝑧 ) ) ∧ ∀ 𝑥 ∈ 𝐵 ( ( ( 𝑥 𝐺 𝑥 ) ‘ ( 1 ‘ 𝑥 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑚 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐷 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ 𝐸 ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑚 ) ) ) ) ) ) |
24 |
4 23
|
mpbid |
⊢ ( 𝜑 → ( 𝐹 : 𝐵 ⟶ ( Base ‘ 𝐸 ) ∧ 𝐺 ∈ X 𝑧 ∈ ( 𝐵 × 𝐵 ) ( ( ( 𝐹 ‘ ( 1st ‘ 𝑧 ) ) ( Hom ‘ 𝐸 ) ( 𝐹 ‘ ( 2nd ‘ 𝑧 ) ) ) ↑m ( ( Hom ‘ 𝐷 ) ‘ 𝑧 ) ) ∧ ∀ 𝑥 ∈ 𝐵 ( ( ( 𝑥 𝐺 𝑥 ) ‘ ( 1 ‘ 𝑥 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑚 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐷 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ 𝐸 ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑚 ) ) ) ) ) |
25 |
24
|
simp3d |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ( ( ( 𝑥 𝐺 𝑥 ) ‘ ( 1 ‘ 𝑥 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑚 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐷 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ 𝐸 ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑚 ) ) ) ) |
26 |
|
simpl |
⊢ ( ( ( ( 𝑥 𝐺 𝑥 ) ‘ ( 1 ‘ 𝑥 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑚 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐷 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ 𝐸 ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑚 ) ) ) → ( ( 𝑥 𝐺 𝑥 ) ‘ ( 1 ‘ 𝑥 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 𝑥 ) ) ) |
27 |
26
|
ralimi |
⊢ ( ∀ 𝑥 ∈ 𝐵 ( ( ( 𝑥 𝐺 𝑥 ) ‘ ( 1 ‘ 𝑥 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ∀ 𝑚 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑛 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑥 𝐺 𝑧 ) ‘ ( 𝑛 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝐷 ) 𝑧 ) 𝑚 ) ) = ( ( ( 𝑦 𝐺 𝑧 ) ‘ 𝑛 ) ( 〈 ( 𝐹 ‘ 𝑥 ) , ( 𝐹 ‘ 𝑦 ) 〉 ( comp ‘ 𝐸 ) ( 𝐹 ‘ 𝑧 ) ) ( ( 𝑥 𝐺 𝑦 ) ‘ 𝑚 ) ) ) → ∀ 𝑥 ∈ 𝐵 ( ( 𝑥 𝐺 𝑥 ) ‘ ( 1 ‘ 𝑥 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 𝑥 ) ) ) |
28 |
25 27
|
syl |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ( ( 𝑥 𝐺 𝑥 ) ‘ ( 1 ‘ 𝑥 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 𝑥 ) ) ) |
29 |
11 28 5
|
rspcdva |
⊢ ( 𝜑 → ( ( 𝑋 𝐺 𝑋 ) ‘ ( 1 ‘ 𝑋 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 𝑋 ) ) ) |