Step |
Hyp |
Ref |
Expression |
1 |
|
rngccat.c |
⊢ 𝐶 = ( RngCat ‘ 𝑈 ) |
2 |
|
rngcid.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
3 |
|
rngcid.o |
⊢ 1 = ( Id ‘ 𝐶 ) |
4 |
|
rngcid.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
5 |
|
rngcid.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
rngcid.s |
⊢ 𝑆 = ( Base ‘ 𝑋 ) |
7 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑈 ∩ Rng ) = ( 𝑈 ∩ Rng ) ) |
8 |
|
eqidd |
⊢ ( 𝜑 → ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) = ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) |
9 |
1 4 7 8
|
rngcval |
⊢ ( 𝜑 → 𝐶 = ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) ) |
10 |
9
|
fveq2d |
⊢ ( 𝜑 → ( Id ‘ 𝐶 ) = ( Id ‘ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) ) ) |
11 |
3 10
|
syl5eq |
⊢ ( 𝜑 → 1 = ( Id ‘ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) ) ) |
12 |
11
|
fveq1d |
⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) = ( ( Id ‘ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) ) ‘ 𝑋 ) ) |
13 |
|
eqid |
⊢ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) = ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) |
14 |
|
eqid |
⊢ ( ExtStrCat ‘ 𝑈 ) = ( ExtStrCat ‘ 𝑈 ) |
15 |
|
incom |
⊢ ( 𝑈 ∩ Rng ) = ( Rng ∩ 𝑈 ) |
16 |
15
|
a1i |
⊢ ( 𝜑 → ( 𝑈 ∩ Rng ) = ( Rng ∩ 𝑈 ) ) |
17 |
14 4 16 8
|
rnghmsubcsetc |
⊢ ( 𝜑 → ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ∈ ( Subcat ‘ ( ExtStrCat ‘ 𝑈 ) ) ) |
18 |
7 8
|
rnghmresfn |
⊢ ( 𝜑 → ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) Fn ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) |
19 |
|
eqid |
⊢ ( Id ‘ ( ExtStrCat ‘ 𝑈 ) ) = ( Id ‘ ( ExtStrCat ‘ 𝑈 ) ) |
20 |
1 2 4
|
rngcbas |
⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Rng ) ) |
21 |
20
|
eleq2d |
⊢ ( 𝜑 → ( 𝑋 ∈ 𝐵 ↔ 𝑋 ∈ ( 𝑈 ∩ Rng ) ) ) |
22 |
5 21
|
mpbid |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑈 ∩ Rng ) ) |
23 |
13 17 18 19 22
|
subcid |
⊢ ( 𝜑 → ( ( Id ‘ ( ExtStrCat ‘ 𝑈 ) ) ‘ 𝑋 ) = ( ( Id ‘ ( ( ExtStrCat ‘ 𝑈 ) ↾cat ( RngHomo ↾ ( ( 𝑈 ∩ Rng ) × ( 𝑈 ∩ Rng ) ) ) ) ) ‘ 𝑋 ) ) |
24 |
|
elinel1 |
⊢ ( 𝑋 ∈ ( 𝑈 ∩ Rng ) → 𝑋 ∈ 𝑈 ) |
25 |
21 24
|
syl6bi |
⊢ ( 𝜑 → ( 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝑈 ) ) |
26 |
5 25
|
mpd |
⊢ ( 𝜑 → 𝑋 ∈ 𝑈 ) |
27 |
14 19 4 26
|
estrcid |
⊢ ( 𝜑 → ( ( Id ‘ ( ExtStrCat ‘ 𝑈 ) ) ‘ 𝑋 ) = ( I ↾ ( Base ‘ 𝑋 ) ) ) |
28 |
6
|
eqcomi |
⊢ ( Base ‘ 𝑋 ) = 𝑆 |
29 |
28
|
a1i |
⊢ ( 𝜑 → ( Base ‘ 𝑋 ) = 𝑆 ) |
30 |
29
|
reseq2d |
⊢ ( 𝜑 → ( I ↾ ( Base ‘ 𝑋 ) ) = ( I ↾ 𝑆 ) ) |
31 |
27 30
|
eqtrd |
⊢ ( 𝜑 → ( ( Id ‘ ( ExtStrCat ‘ 𝑈 ) ) ‘ 𝑋 ) = ( I ↾ 𝑆 ) ) |
32 |
12 23 31
|
3eqtr2d |
⊢ ( 𝜑 → ( 1 ‘ 𝑋 ) = ( I ↾ 𝑆 ) ) |