Step |
Hyp |
Ref |
Expression |
1 |
|
catidcl.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
2 |
|
catidcl.h |
⊢ 𝐻 = ( Hom ‘ 𝐶 ) |
3 |
|
catidcl.i |
⊢ 1 = ( Id ‘ 𝐶 ) |
4 |
|
catidcl.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
5 |
|
catidcl.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
catlid.o |
⊢ · = ( comp ‘ 𝐶 ) |
7 |
|
catlid.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
8 |
|
catlid.f |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ) |
9 |
|
oveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑌 ) ( 1 ‘ 𝑋 ) ) = ( 𝐹 ( 〈 𝑋 , 𝑋 〉 · 𝑌 ) ( 1 ‘ 𝑋 ) ) ) |
10 |
|
id |
⊢ ( 𝑓 = 𝐹 → 𝑓 = 𝐹 ) |
11 |
9 10
|
eqeq12d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑌 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ↔ ( 𝐹 ( 〈 𝑋 , 𝑋 〉 · 𝑌 ) ( 1 ‘ 𝑋 ) ) = 𝐹 ) ) |
12 |
|
oveq2 |
⊢ ( 𝑦 = 𝑌 → ( 𝑋 𝐻 𝑦 ) = ( 𝑋 𝐻 𝑌 ) ) |
13 |
|
oveq2 |
⊢ ( 𝑦 = 𝑌 → ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) = ( 〈 𝑋 , 𝑋 〉 · 𝑌 ) ) |
14 |
13
|
oveqd |
⊢ ( 𝑦 = 𝑌 → ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) ( 1 ‘ 𝑋 ) ) = ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑌 ) ( 1 ‘ 𝑋 ) ) ) |
15 |
14
|
eqeq1d |
⊢ ( 𝑦 = 𝑌 → ( ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ↔ ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑌 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ) ) |
16 |
12 15
|
raleqbidv |
⊢ ( 𝑦 = 𝑌 → ( ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ↔ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑌 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ) ) |
17 |
|
simpr |
⊢ ( ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( 〈 𝑦 , 𝑋 〉 · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ) → ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ) |
18 |
17
|
ralimi |
⊢ ( ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( 〈 𝑦 , 𝑋 〉 · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ) → ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ) |
19 |
18
|
a1i |
⊢ ( 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) → ( ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( 〈 𝑦 , 𝑋 〉 · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ) → ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ) ) |
20 |
19
|
ss2rabi |
⊢ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( 〈 𝑦 , 𝑋 〉 · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ) } ⊆ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 } |
21 |
1 2 6 4 3 5
|
cidval |
⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) = ( ℩ 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( 〈 𝑦 , 𝑋 〉 · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ) ) ) |
22 |
1 2 6 4 5
|
catideu |
⊢ ( 𝜑 → ∃! 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( 〈 𝑦 , 𝑋 〉 · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ) ) |
23 |
|
riotacl2 |
⊢ ( ∃! 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( 〈 𝑦 , 𝑋 〉 · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ) → ( ℩ 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( 〈 𝑦 , 𝑋 〉 · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ) ) ∈ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( 〈 𝑦 , 𝑋 〉 · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ) } ) |
24 |
22 23
|
syl |
⊢ ( 𝜑 → ( ℩ 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( 〈 𝑦 , 𝑋 〉 · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ) ) ∈ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( 〈 𝑦 , 𝑋 〉 · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ) } ) |
25 |
21 24
|
eqeltrd |
⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) ∈ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ( ∀ 𝑓 ∈ ( 𝑦 𝐻 𝑋 ) ( 𝑔 ( 〈 𝑦 , 𝑋 〉 · 𝑋 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ) } ) |
26 |
20 25
|
sselid |
⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) ∈ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 } ) |
27 |
|
oveq2 |
⊢ ( 𝑔 = ( 1 ‘ 𝑋 ) → ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) ( 1 ‘ 𝑋 ) ) ) |
28 |
27
|
eqeq1d |
⊢ ( 𝑔 = ( 1 ‘ 𝑋 ) → ( ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ↔ ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ) ) |
29 |
28
|
2ralbidv |
⊢ ( 𝑔 = ( 1 ‘ 𝑋 ) → ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 ↔ ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ) ) |
30 |
29
|
elrab |
⊢ ( ( 1 ‘ 𝑋 ) ∈ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 } ↔ ( ( 1 ‘ 𝑋 ) ∈ ( 𝑋 𝐻 𝑋 ) ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ) ) |
31 |
30
|
simprbi |
⊢ ( ( 1 ‘ 𝑋 ) ∈ { 𝑔 ∈ ( 𝑋 𝐻 𝑋 ) ∣ ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) 𝑔 ) = 𝑓 } → ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ) |
32 |
26 31
|
syl |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐵 ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑦 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑦 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ) |
33 |
16 32 7
|
rspcdva |
⊢ ( 𝜑 → ∀ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ( 𝑓 ( 〈 𝑋 , 𝑋 〉 · 𝑌 ) ( 1 ‘ 𝑋 ) ) = 𝑓 ) |
34 |
11 33 8
|
rspcdva |
⊢ ( 𝜑 → ( 𝐹 ( 〈 𝑋 , 𝑋 〉 · 𝑌 ) ( 1 ‘ 𝑋 ) ) = 𝐹 ) |