Step |
Hyp |
Ref |
Expression |
1 |
|
rngcval.c |
⊢ 𝐶 = ( RngCat ‘ 𝑈 ) |
2 |
|
rngcval.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
3 |
|
rngcval.b |
⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Rng ) ) |
4 |
|
rngcval.h |
⊢ ( 𝜑 → 𝐻 = ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) ) |
5 |
|
df-rngc |
⊢ RngCat = ( 𝑢 ∈ V ↦ ( ( ExtStrCat ‘ 𝑢 ) ↾cat ( RngHomo ↾ ( ( 𝑢 ∩ Rng ) × ( 𝑢 ∩ Rng ) ) ) ) ) |
6 |
|
fveq2 |
⊢ ( 𝑢 = 𝑈 → ( ExtStrCat ‘ 𝑢 ) = ( ExtStrCat ‘ 𝑈 ) ) |
7 |
6
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑢 = 𝑈 ) → ( ExtStrCat ‘ 𝑢 ) = ( ExtStrCat ‘ 𝑈 ) ) |
8 |
|
ineq1 |
⊢ ( 𝑢 = 𝑈 → ( 𝑢 ∩ Rng ) = ( 𝑈 ∩ Rng ) ) |
9 |
8
|
sqxpeqd |
⊢ ( 𝑢 = 𝑈 → ( ( 𝑢 ∩ Rng ) × ( 𝑢 ∩ Rng ) ) = ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) |
10 |
3
|
sqxpeqd |
⊢ ( 𝜑 → ( 𝐵 × 𝐵 ) = ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) |
11 |
10
|
eqcomd |
⊢ ( 𝜑 → ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) = ( 𝐵 × 𝐵 ) ) |
12 |
9 11
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ 𝑢 = 𝑈 ) → ( ( 𝑢 ∩ Rng ) × ( 𝑢 ∩ Rng ) ) = ( 𝐵 × 𝐵 ) ) |
13 |
12
|
reseq2d |
⊢ ( ( 𝜑 ∧ 𝑢 = 𝑈 ) → ( RngHomo ↾ ( ( 𝑢 ∩ Rng ) × ( 𝑢 ∩ Rng ) ) ) = ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) ) |
14 |
4
|
eqcomd |
⊢ ( 𝜑 → ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) = 𝐻 ) |
15 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑢 = 𝑈 ) → ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) = 𝐻 ) |
16 |
13 15
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑢 = 𝑈 ) → ( RngHomo ↾ ( ( 𝑢 ∩ Rng ) × ( 𝑢 ∩ Rng ) ) ) = 𝐻 ) |
17 |
7 16
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑢 = 𝑈 ) → ( ( ExtStrCat ‘ 𝑢 ) ↾cat ( RngHomo ↾ ( ( 𝑢 ∩ Rng ) × ( 𝑢 ∩ Rng ) ) ) ) = ( ( ExtStrCat ‘ 𝑈 ) ↾cat 𝐻 ) ) |
18 |
2
|
elexd |
⊢ ( 𝜑 → 𝑈 ∈ V ) |
19 |
|
ovexd |
⊢ ( 𝜑 → ( ( ExtStrCat ‘ 𝑈 ) ↾cat 𝐻 ) ∈ V ) |
20 |
5 17 18 19
|
fvmptd2 |
⊢ ( 𝜑 → ( RngCat ‘ 𝑈 ) = ( ( ExtStrCat ‘ 𝑈 ) ↾cat 𝐻 ) ) |
21 |
1 20
|
syl5eq |
⊢ ( 𝜑 → 𝐶 = ( ( ExtStrCat ‘ 𝑈 ) ↾cat 𝐻 ) ) |