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 |
|
funcringcsetcALTV.g |
⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) ) |
8 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) |
9 |
|
ovex |
⊢ ( 𝑥 RingHom 𝑦 ) ∈ V |
10 |
|
id |
⊢ ( ( 𝑥 RingHom 𝑦 ) ∈ V → ( 𝑥 RingHom 𝑦 ) ∈ V ) |
11 |
10
|
resiexd |
⊢ ( ( 𝑥 RingHom 𝑦 ) ∈ V → ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ∈ V ) |
12 |
9 11
|
ax-mp |
⊢ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ∈ V |
13 |
8 12
|
fnmpoi |
⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) Fn ( 𝐵 × 𝐵 ) |
14 |
7
|
fneq1d |
⊢ ( 𝜑 → ( 𝐺 Fn ( 𝐵 × 𝐵 ) ↔ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) Fn ( 𝐵 × 𝐵 ) ) ) |
15 |
13 14
|
mpbiri |
⊢ ( 𝜑 → 𝐺 Fn ( 𝐵 × 𝐵 ) ) |