Step |
Hyp |
Ref |
Expression |
1 |
|
rngcrescrhm.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
2 |
|
rngcrescrhm.c |
⊢ 𝐶 = ( RngCat ‘ 𝑈 ) |
3 |
|
rngcrescrhm.r |
⊢ ( 𝜑 → 𝑅 = ( Ring ∩ 𝑈 ) ) |
4 |
|
rngcrescrhm.h |
⊢ 𝐻 = ( RingHom ↾ ( 𝑅 × 𝑅 ) ) |
5 |
|
eqid |
⊢ ( 𝐶 ↾cat 𝐻 ) = ( 𝐶 ↾cat 𝐻 ) |
6 |
2
|
fvexi |
⊢ 𝐶 ∈ V |
7 |
6
|
a1i |
⊢ ( 𝜑 → 𝐶 ∈ V ) |
8 |
|
incom |
⊢ ( Ring ∩ 𝑈 ) = ( 𝑈 ∩ Ring ) |
9 |
3 8
|
eqtrdi |
⊢ ( 𝜑 → 𝑅 = ( 𝑈 ∩ Ring ) ) |
10 |
|
inex1g |
⊢ ( 𝑈 ∈ 𝑉 → ( 𝑈 ∩ Ring ) ∈ V ) |
11 |
1 10
|
syl |
⊢ ( 𝜑 → ( 𝑈 ∩ Ring ) ∈ V ) |
12 |
9 11
|
eqeltrd |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
13 |
|
inss1 |
⊢ ( Ring ∩ 𝑈 ) ⊆ Ring |
14 |
3 13
|
eqsstrdi |
⊢ ( 𝜑 → 𝑅 ⊆ Ring ) |
15 |
|
xpss12 |
⊢ ( ( 𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring ) → ( 𝑅 × 𝑅 ) ⊆ ( Ring × Ring ) ) |
16 |
14 14 15
|
syl2anc |
⊢ ( 𝜑 → ( 𝑅 × 𝑅 ) ⊆ ( Ring × Ring ) ) |
17 |
|
rhmfn |
⊢ RingHom Fn ( Ring × Ring ) |
18 |
|
fnssresb |
⊢ ( RingHom Fn ( Ring × Ring ) → ( ( RingHom ↾ ( 𝑅 × 𝑅 ) ) Fn ( 𝑅 × 𝑅 ) ↔ ( 𝑅 × 𝑅 ) ⊆ ( Ring × Ring ) ) ) |
19 |
17 18
|
mp1i |
⊢ ( 𝜑 → ( ( RingHom ↾ ( 𝑅 × 𝑅 ) ) Fn ( 𝑅 × 𝑅 ) ↔ ( 𝑅 × 𝑅 ) ⊆ ( Ring × Ring ) ) ) |
20 |
16 19
|
mpbird |
⊢ ( 𝜑 → ( RingHom ↾ ( 𝑅 × 𝑅 ) ) Fn ( 𝑅 × 𝑅 ) ) |
21 |
4
|
fneq1i |
⊢ ( 𝐻 Fn ( 𝑅 × 𝑅 ) ↔ ( RingHom ↾ ( 𝑅 × 𝑅 ) ) Fn ( 𝑅 × 𝑅 ) ) |
22 |
20 21
|
sylibr |
⊢ ( 𝜑 → 𝐻 Fn ( 𝑅 × 𝑅 ) ) |
23 |
5 7 12 22
|
rescval2 |
⊢ ( 𝜑 → ( 𝐶 ↾cat 𝐻 ) = ( ( 𝐶 ↾s 𝑅 ) sSet 〈 ( Hom ‘ ndx ) , 𝐻 〉 ) ) |