Step |
Hyp |
Ref |
Expression |
1 |
|
issect.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
2 |
|
issect.h |
⊢ 𝐻 = ( Hom ‘ 𝐶 ) |
3 |
|
issect.o |
⊢ · = ( comp ‘ 𝐶 ) |
4 |
|
issect.i |
⊢ 1 = ( Id ‘ 𝐶 ) |
5 |
|
issect.s |
⊢ 𝑆 = ( Sect ‘ 𝐶 ) |
6 |
|
issect.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
7 |
|
issect.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
8 |
|
issect.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
9 |
1 2 3 4 5 6 7 7
|
sectffval |
⊢ ( 𝜑 → 𝑆 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑥 ) ) ∧ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑥 ) 𝑓 ) = ( 1 ‘ 𝑥 ) ) } ) ) |
10 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 𝑥 = 𝑋 ) |
11 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 𝑦 = 𝑌 ) |
12 |
10 11
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑥 𝐻 𝑦 ) = ( 𝑋 𝐻 𝑌 ) ) |
13 |
12
|
eleq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ↔ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ) ) |
14 |
11 10
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑦 𝐻 𝑥 ) = ( 𝑌 𝐻 𝑋 ) ) |
15 |
14
|
eleq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑔 ∈ ( 𝑦 𝐻 𝑥 ) ↔ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ) |
16 |
13 15
|
anbi12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑥 ) ) ↔ ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ) ) |
17 |
10 11
|
opeq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 〈 𝑥 , 𝑦 〉 = 〈 𝑋 , 𝑌 〉 ) |
18 |
17 10
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 〈 𝑥 , 𝑦 〉 · 𝑥 ) = ( 〈 𝑋 , 𝑌 〉 · 𝑋 ) ) |
19 |
18
|
oveqd |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑥 ) 𝑓 ) = ( 𝑔 ( 〈 𝑋 , 𝑌 〉 · 𝑋 ) 𝑓 ) ) |
20 |
10
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 1 ‘ 𝑥 ) = ( 1 ‘ 𝑋 ) ) |
21 |
19 20
|
eqeq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑥 ) 𝑓 ) = ( 1 ‘ 𝑥 ) ↔ ( 𝑔 ( 〈 𝑋 , 𝑌 〉 · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) ) |
22 |
16 21
|
anbi12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( ( ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑥 ) ) ∧ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑥 ) 𝑓 ) = ( 1 ‘ 𝑥 ) ) ↔ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( 〈 𝑋 , 𝑌 〉 · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) ) ) |
23 |
22
|
opabbidv |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∧ 𝑔 ∈ ( 𝑦 𝐻 𝑥 ) ) ∧ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑥 ) 𝑓 ) = ( 1 ‘ 𝑥 ) ) } = { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( 〈 𝑋 , 𝑌 〉 · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } ) |
24 |
|
ovex |
⊢ ( 𝑋 𝐻 𝑌 ) ∈ V |
25 |
|
ovex |
⊢ ( 𝑌 𝐻 𝑋 ) ∈ V |
26 |
24 25
|
xpex |
⊢ ( ( 𝑋 𝐻 𝑌 ) × ( 𝑌 𝐻 𝑋 ) ) ∈ V |
27 |
|
opabssxp |
⊢ { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( 〈 𝑋 , 𝑌 〉 · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } ⊆ ( ( 𝑋 𝐻 𝑌 ) × ( 𝑌 𝐻 𝑋 ) ) |
28 |
26 27
|
ssexi |
⊢ { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( 〈 𝑋 , 𝑌 〉 · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } ∈ V |
29 |
28
|
a1i |
⊢ ( 𝜑 → { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( 〈 𝑋 , 𝑌 〉 · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } ∈ V ) |
30 |
9 23 7 8 29
|
ovmpod |
⊢ ( 𝜑 → ( 𝑋 𝑆 𝑌 ) = { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( 〈 𝑋 , 𝑌 〉 · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } ) |