Step |
Hyp |
Ref |
Expression |
1 |
|
yoneda.y |
⊢ 𝑌 = ( Yon ‘ 𝐶 ) |
2 |
|
yoneda.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
3 |
|
yoneda.1 |
⊢ 1 = ( Id ‘ 𝐶 ) |
4 |
|
yoneda.o |
⊢ 𝑂 = ( oppCat ‘ 𝐶 ) |
5 |
|
yoneda.s |
⊢ 𝑆 = ( SetCat ‘ 𝑈 ) |
6 |
|
yoneda.t |
⊢ 𝑇 = ( SetCat ‘ 𝑉 ) |
7 |
|
yoneda.q |
⊢ 𝑄 = ( 𝑂 FuncCat 𝑆 ) |
8 |
|
yoneda.h |
⊢ 𝐻 = ( HomF ‘ 𝑄 ) |
9 |
|
yoneda.r |
⊢ 𝑅 = ( ( 𝑄 ×c 𝑂 ) FuncCat 𝑇 ) |
10 |
|
yoneda.e |
⊢ 𝐸 = ( 𝑂 evalF 𝑆 ) |
11 |
|
yoneda.z |
⊢ 𝑍 = ( 𝐻 ∘func ( ( 〈 ( 1st ‘ 𝑌 ) , tpos ( 2nd ‘ 𝑌 ) 〉 ∘func ( 𝑄 2ndF 𝑂 ) ) 〈,〉F ( 𝑄 1stF 𝑂 ) ) ) |
12 |
|
yoneda.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
13 |
|
yoneda.w |
⊢ ( 𝜑 → 𝑉 ∈ 𝑊 ) |
14 |
|
yoneda.u |
⊢ ( 𝜑 → ran ( Homf ‘ 𝐶 ) ⊆ 𝑈 ) |
15 |
|
yoneda.v |
⊢ ( 𝜑 → ( ran ( Homf ‘ 𝑄 ) ∪ 𝑈 ) ⊆ 𝑉 ) |
16 |
|
yoneda.m |
⊢ 𝑀 = ( 𝑓 ∈ ( 𝑂 Func 𝑆 ) , 𝑥 ∈ 𝐵 ↦ ( 𝑎 ∈ ( ( ( 1st ‘ 𝑌 ) ‘ 𝑥 ) ( 𝑂 Nat 𝑆 ) 𝑓 ) ↦ ( ( 𝑎 ‘ 𝑥 ) ‘ ( 1 ‘ 𝑥 ) ) ) ) |
17 |
|
yoneda.i |
⊢ 𝐼 = ( Iso ‘ 𝑅 ) |
18 |
9
|
fucbas |
⊢ ( ( 𝑄 ×c 𝑂 ) Func 𝑇 ) = ( Base ‘ 𝑅 ) |
19 |
|
eqid |
⊢ ( Inv ‘ 𝑅 ) = ( Inv ‘ 𝑅 ) |
20 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
|
yonedalem1 |
⊢ ( 𝜑 → ( 𝑍 ∈ ( ( 𝑄 ×c 𝑂 ) Func 𝑇 ) ∧ 𝐸 ∈ ( ( 𝑄 ×c 𝑂 ) Func 𝑇 ) ) ) |
21 |
20
|
simpld |
⊢ ( 𝜑 → 𝑍 ∈ ( ( 𝑄 ×c 𝑂 ) Func 𝑇 ) ) |
22 |
|
funcrcl |
⊢ ( 𝑍 ∈ ( ( 𝑄 ×c 𝑂 ) Func 𝑇 ) → ( ( 𝑄 ×c 𝑂 ) ∈ Cat ∧ 𝑇 ∈ Cat ) ) |
23 |
21 22
|
syl |
⊢ ( 𝜑 → ( ( 𝑄 ×c 𝑂 ) ∈ Cat ∧ 𝑇 ∈ Cat ) ) |
24 |
23
|
simpld |
⊢ ( 𝜑 → ( 𝑄 ×c 𝑂 ) ∈ Cat ) |
25 |
23
|
simprd |
⊢ ( 𝜑 → 𝑇 ∈ Cat ) |
26 |
9 24 25
|
fuccat |
⊢ ( 𝜑 → 𝑅 ∈ Cat ) |
27 |
20
|
simprd |
⊢ ( 𝜑 → 𝐸 ∈ ( ( 𝑄 ×c 𝑂 ) Func 𝑇 ) ) |
28 |
|
eqid |
⊢ ( 𝑓 ∈ ( 𝑂 Func 𝑆 ) , 𝑥 ∈ 𝐵 ↦ ( 𝑢 ∈ ( ( 1st ‘ 𝑓 ) ‘ 𝑥 ) ↦ ( 𝑦 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ↦ ( ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ 𝑔 ) ‘ 𝑢 ) ) ) ) ) = ( 𝑓 ∈ ( 𝑂 Func 𝑆 ) , 𝑥 ∈ 𝐵 ↦ ( 𝑢 ∈ ( ( 1st ‘ 𝑓 ) ‘ 𝑥 ) ↦ ( 𝑦 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ↦ ( ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ 𝑔 ) ‘ 𝑢 ) ) ) ) ) |
29 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 19 28
|
yonedainv |
⊢ ( 𝜑 → 𝑀 ( 𝑍 ( Inv ‘ 𝑅 ) 𝐸 ) ( 𝑓 ∈ ( 𝑂 Func 𝑆 ) , 𝑥 ∈ 𝐵 ↦ ( 𝑢 ∈ ( ( 1st ‘ 𝑓 ) ‘ 𝑥 ) ↦ ( 𝑦 ∈ 𝐵 ↦ ( 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ↦ ( ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ 𝑔 ) ‘ 𝑢 ) ) ) ) ) ) |
30 |
18 19 26 21 27 17 29
|
inviso1 |
⊢ ( 𝜑 → 𝑀 ∈ ( 𝑍 𝐼 𝐸 ) ) |