Step |
Hyp |
Ref |
Expression |
1 |
|
isohom.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
2 |
|
isohom.h |
⊢ 𝐻 = ( Hom ‘ 𝐶 ) |
3 |
|
isohom.i |
⊢ 𝐼 = ( Iso ‘ 𝐶 ) |
4 |
|
isohom.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
5 |
|
isohom.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
isohom.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
7 |
|
eqid |
⊢ ( Inv ‘ 𝐶 ) = ( Inv ‘ 𝐶 ) |
8 |
1 7 4 5 6 3
|
isoval |
⊢ ( 𝜑 → ( 𝑋 𝐼 𝑌 ) = dom ( 𝑋 ( Inv ‘ 𝐶 ) 𝑌 ) ) |
9 |
1 7 4 5 6 2
|
invss |
⊢ ( 𝜑 → ( 𝑋 ( Inv ‘ 𝐶 ) 𝑌 ) ⊆ ( ( 𝑋 𝐻 𝑌 ) × ( 𝑌 𝐻 𝑋 ) ) ) |
10 |
|
dmss |
⊢ ( ( 𝑋 ( Inv ‘ 𝐶 ) 𝑌 ) ⊆ ( ( 𝑋 𝐻 𝑌 ) × ( 𝑌 𝐻 𝑋 ) ) → dom ( 𝑋 ( Inv ‘ 𝐶 ) 𝑌 ) ⊆ dom ( ( 𝑋 𝐻 𝑌 ) × ( 𝑌 𝐻 𝑋 ) ) ) |
11 |
9 10
|
syl |
⊢ ( 𝜑 → dom ( 𝑋 ( Inv ‘ 𝐶 ) 𝑌 ) ⊆ dom ( ( 𝑋 𝐻 𝑌 ) × ( 𝑌 𝐻 𝑋 ) ) ) |
12 |
8 11
|
eqsstrd |
⊢ ( 𝜑 → ( 𝑋 𝐼 𝑌 ) ⊆ dom ( ( 𝑋 𝐻 𝑌 ) × ( 𝑌 𝐻 𝑋 ) ) ) |
13 |
|
dmxpss |
⊢ dom ( ( 𝑋 𝐻 𝑌 ) × ( 𝑌 𝐻 𝑋 ) ) ⊆ ( 𝑋 𝐻 𝑌 ) |
14 |
12 13
|
sstrdi |
⊢ ( 𝜑 → ( 𝑋 𝐼 𝑌 ) ⊆ ( 𝑋 𝐻 𝑌 ) ) |