Step |
Hyp |
Ref |
Expression |
1 |
|
funcringcsetcALTV.r |
⊢ 𝑅 = ( RingCatALTV ‘ 𝑈 ) |
2 |
|
funcringcsetcALTV.s |
⊢ 𝑆 = ( SetCat ‘ 𝑈 ) |
3 |
|
funcringcsetcALTV.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
4 |
|
funcringcsetcALTV.c |
⊢ 𝐶 = ( Base ‘ 𝑆 ) |
5 |
|
funcringcsetcALTV.u |
⊢ ( 𝜑 → 𝑈 ∈ WUni ) |
6 |
|
funcringcsetcALTV.f |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) ) |
7 |
1 3 5
|
ringcbasbasALTV |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( Base ‘ 𝑥 ) ∈ 𝑈 ) |
8 |
2 5
|
setcbas |
⊢ ( 𝜑 → 𝑈 = ( Base ‘ 𝑆 ) ) |
9 |
8
|
eqcomd |
⊢ ( 𝜑 → ( Base ‘ 𝑆 ) = 𝑈 ) |
10 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( Base ‘ 𝑆 ) = 𝑈 ) |
11 |
7 10
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( Base ‘ 𝑥 ) ∈ ( Base ‘ 𝑆 ) ) |
12 |
11 4
|
eleqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( Base ‘ 𝑥 ) ∈ 𝐶 ) |
13 |
12
|
fmpttd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) : 𝐵 ⟶ 𝐶 ) |
14 |
6
|
feq1d |
⊢ ( 𝜑 → ( 𝐹 : 𝐵 ⟶ 𝐶 ↔ ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) : 𝐵 ⟶ 𝐶 ) ) |
15 |
13 14
|
mpbird |
⊢ ( 𝜑 → 𝐹 : 𝐵 ⟶ 𝐶 ) |