Step |
Hyp |
Ref |
Expression |
1 |
|
1stfval.t |
⊢ 𝑇 = ( 𝐶 ×c 𝐷 ) |
2 |
|
1stfval.b |
⊢ 𝐵 = ( Base ‘ 𝑇 ) |
3 |
|
1stfval.h |
⊢ 𝐻 = ( Hom ‘ 𝑇 ) |
4 |
|
1stfval.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
5 |
|
1stfval.d |
⊢ ( 𝜑 → 𝐷 ∈ Cat ) |
6 |
|
2ndfval.p |
⊢ 𝑄 = ( 𝐶 2ndF 𝐷 ) |
7 |
|
2ndf1.p |
⊢ ( 𝜑 → 𝑅 ∈ 𝐵 ) |
8 |
|
2ndf2.p |
⊢ ( 𝜑 → 𝑆 ∈ 𝐵 ) |
9 |
1 2 3 4 5 6
|
2ndfval |
⊢ ( 𝜑 → 𝑄 = 〈 ( 2nd ↾ 𝐵 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 2nd ↾ ( 𝑥 𝐻 𝑦 ) ) ) 〉 ) |
10 |
|
fo2nd |
⊢ 2nd : V –onto→ V |
11 |
|
fofun |
⊢ ( 2nd : V –onto→ V → Fun 2nd ) |
12 |
10 11
|
ax-mp |
⊢ Fun 2nd |
13 |
2
|
fvexi |
⊢ 𝐵 ∈ V |
14 |
|
resfunexg |
⊢ ( ( Fun 2nd ∧ 𝐵 ∈ V ) → ( 2nd ↾ 𝐵 ) ∈ V ) |
15 |
12 13 14
|
mp2an |
⊢ ( 2nd ↾ 𝐵 ) ∈ V |
16 |
13 13
|
mpoex |
⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 2nd ↾ ( 𝑥 𝐻 𝑦 ) ) ) ∈ V |
17 |
15 16
|
op2ndd |
⊢ ( 𝑄 = 〈 ( 2nd ↾ 𝐵 ) , ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 2nd ↾ ( 𝑥 𝐻 𝑦 ) ) ) 〉 → ( 2nd ‘ 𝑄 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 2nd ↾ ( 𝑥 𝐻 𝑦 ) ) ) ) |
18 |
9 17
|
syl |
⊢ ( 𝜑 → ( 2nd ‘ 𝑄 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 2nd ↾ ( 𝑥 𝐻 𝑦 ) ) ) ) |
19 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑅 ∧ 𝑦 = 𝑆 ) ) → 𝑥 = 𝑅 ) |
20 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑅 ∧ 𝑦 = 𝑆 ) ) → 𝑦 = 𝑆 ) |
21 |
19 20
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑅 ∧ 𝑦 = 𝑆 ) ) → ( 𝑥 𝐻 𝑦 ) = ( 𝑅 𝐻 𝑆 ) ) |
22 |
21
|
reseq2d |
⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑅 ∧ 𝑦 = 𝑆 ) ) → ( 2nd ↾ ( 𝑥 𝐻 𝑦 ) ) = ( 2nd ↾ ( 𝑅 𝐻 𝑆 ) ) ) |
23 |
|
ovex |
⊢ ( 𝑅 𝐻 𝑆 ) ∈ V |
24 |
|
resfunexg |
⊢ ( ( Fun 2nd ∧ ( 𝑅 𝐻 𝑆 ) ∈ V ) → ( 2nd ↾ ( 𝑅 𝐻 𝑆 ) ) ∈ V ) |
25 |
12 23 24
|
mp2an |
⊢ ( 2nd ↾ ( 𝑅 𝐻 𝑆 ) ) ∈ V |
26 |
25
|
a1i |
⊢ ( 𝜑 → ( 2nd ↾ ( 𝑅 𝐻 𝑆 ) ) ∈ V ) |
27 |
18 22 7 8 26
|
ovmpod |
⊢ ( 𝜑 → ( 𝑅 ( 2nd ‘ 𝑄 ) 𝑆 ) = ( 2nd ↾ ( 𝑅 𝐻 𝑆 ) ) ) |