Step |
Hyp |
Ref |
Expression |
1 |
|
oyoncl.o |
⊢ 𝑂 = ( oppCat ‘ 𝐶 ) |
2 |
|
oyoncl.y |
⊢ 𝑌 = ( Yon ‘ 𝑂 ) |
3 |
|
oyoncl.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
4 |
|
oyoncl.s |
⊢ 𝑆 = ( SetCat ‘ 𝑈 ) |
5 |
|
oyoncl.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
6 |
|
oyoncl.h |
⊢ ( 𝜑 → ran ( Homf ‘ 𝐶 ) ⊆ 𝑈 ) |
7 |
|
oyoncl.q |
⊢ 𝑄 = ( 𝐶 FuncCat 𝑆 ) |
8 |
1
|
oppccat |
⊢ ( 𝐶 ∈ Cat → 𝑂 ∈ Cat ) |
9 |
3 8
|
syl |
⊢ ( 𝜑 → 𝑂 ∈ Cat ) |
10 |
|
eqid |
⊢ ( oppCat ‘ 𝑂 ) = ( oppCat ‘ 𝑂 ) |
11 |
|
eqid |
⊢ ( ( oppCat ‘ 𝑂 ) FuncCat 𝑆 ) = ( ( oppCat ‘ 𝑂 ) FuncCat 𝑆 ) |
12 |
|
eqid |
⊢ ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐶 ) |
13 |
1 12
|
oppchomf |
⊢ tpos ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝑂 ) |
14 |
13
|
rneqi |
⊢ ran tpos ( Homf ‘ 𝐶 ) = ran ( Homf ‘ 𝑂 ) |
15 |
|
relxp |
⊢ Rel ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) |
16 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
17 |
12 16
|
homffn |
⊢ ( Homf ‘ 𝐶 ) Fn ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) |
18 |
17
|
fndmi |
⊢ dom ( Homf ‘ 𝐶 ) = ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) |
19 |
18
|
releqi |
⊢ ( Rel dom ( Homf ‘ 𝐶 ) ↔ Rel ( ( Base ‘ 𝐶 ) × ( Base ‘ 𝐶 ) ) ) |
20 |
15 19
|
mpbir |
⊢ Rel dom ( Homf ‘ 𝐶 ) |
21 |
|
rntpos |
⊢ ( Rel dom ( Homf ‘ 𝐶 ) → ran tpos ( Homf ‘ 𝐶 ) = ran ( Homf ‘ 𝐶 ) ) |
22 |
20 21
|
ax-mp |
⊢ ran tpos ( Homf ‘ 𝐶 ) = ran ( Homf ‘ 𝐶 ) |
23 |
14 22
|
eqtr3i |
⊢ ran ( Homf ‘ 𝑂 ) = ran ( Homf ‘ 𝐶 ) |
24 |
23 6
|
eqsstrid |
⊢ ( 𝜑 → ran ( Homf ‘ 𝑂 ) ⊆ 𝑈 ) |
25 |
2 9 10 4 11 5 24
|
yoncl |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑂 Func ( ( oppCat ‘ 𝑂 ) FuncCat 𝑆 ) ) ) |
26 |
1
|
2oppchomf |
⊢ ( Homf ‘ 𝐶 ) = ( Homf ‘ ( oppCat ‘ 𝑂 ) ) |
27 |
26
|
a1i |
⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ ( oppCat ‘ 𝑂 ) ) ) |
28 |
1
|
2oppccomf |
⊢ ( compf ‘ 𝐶 ) = ( compf ‘ ( oppCat ‘ 𝑂 ) ) |
29 |
28
|
a1i |
⊢ ( 𝜑 → ( compf ‘ 𝐶 ) = ( compf ‘ ( oppCat ‘ 𝑂 ) ) ) |
30 |
|
eqidd |
⊢ ( 𝜑 → ( Homf ‘ 𝑆 ) = ( Homf ‘ 𝑆 ) ) |
31 |
|
eqidd |
⊢ ( 𝜑 → ( compf ‘ 𝑆 ) = ( compf ‘ 𝑆 ) ) |
32 |
10
|
oppccat |
⊢ ( 𝑂 ∈ Cat → ( oppCat ‘ 𝑂 ) ∈ Cat ) |
33 |
9 32
|
syl |
⊢ ( 𝜑 → ( oppCat ‘ 𝑂 ) ∈ Cat ) |
34 |
4
|
setccat |
⊢ ( 𝑈 ∈ 𝑉 → 𝑆 ∈ Cat ) |
35 |
5 34
|
syl |
⊢ ( 𝜑 → 𝑆 ∈ Cat ) |
36 |
27 29 30 31 3 33 35 35
|
fucpropd |
⊢ ( 𝜑 → ( 𝐶 FuncCat 𝑆 ) = ( ( oppCat ‘ 𝑂 ) FuncCat 𝑆 ) ) |
37 |
7 36
|
eqtrid |
⊢ ( 𝜑 → 𝑄 = ( ( oppCat ‘ 𝑂 ) FuncCat 𝑆 ) ) |
38 |
37
|
oveq2d |
⊢ ( 𝜑 → ( 𝑂 Func 𝑄 ) = ( 𝑂 Func ( ( oppCat ‘ 𝑂 ) FuncCat 𝑆 ) ) ) |
39 |
25 38
|
eleqtrrd |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑂 Func 𝑄 ) ) |