Step |
Hyp |
Ref |
Expression |
1 |
|
rngcbas.c |
⊢ 𝐶 = ( RngCat ‘ 𝑈 ) |
2 |
|
rngcbas.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
3 |
|
rngcbas.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
4 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑈 ∩ Rng ) = ( 𝑈 ∩ Rng ) ) |
5 |
|
eqidd |
⊢ ( 𝜑 → ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) = ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) |
6 |
1 3 4 5
|
rngcval |
⊢ ( 𝜑 → 𝐶 = ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) ) |
7 |
6
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) ) ) |
8 |
2
|
a1i |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐶 ) ) |
9 |
|
eqid |
⊢ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) = ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) |
10 |
|
eqid |
⊢ ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) = ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) |
11 |
|
fvexd |
⊢ ( 𝜑 → ( ExtStrCat ‘ 𝑈 ) ∈ V ) |
12 |
4 5
|
rnghmresfn |
⊢ ( 𝜑 → ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) Fn ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) |
13 |
|
inss1 |
⊢ ( 𝑈 ∩ Rng ) ⊆ 𝑈 |
14 |
|
eqid |
⊢ ( ExtStrCat ‘ 𝑈 ) = ( ExtStrCat ‘ 𝑈 ) |
15 |
14 3
|
estrcbas |
⊢ ( 𝜑 → 𝑈 = ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) ) |
16 |
13 15
|
sseqtrid |
⊢ ( 𝜑 → ( 𝑈 ∩ Rng ) ⊆ ( Base ‘ ( ExtStrCat ‘ 𝑈 ) ) ) |
17 |
9 10 11 12 16
|
rescbas |
⊢ ( 𝜑 → ( 𝑈 ∩ Rng ) = ( Base ‘ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) ) ) |
18 |
7 8 17
|
3eqtr4d |
⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Rng ) ) |