Step |
Hyp |
Ref |
Expression |
1 |
|
invfval.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
2 |
|
invfval.n |
⊢ 𝑁 = ( Inv ‘ 𝐶 ) |
3 |
|
invfval.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
4 |
|
invfval.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
5 |
|
invfval.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
6 |
|
isoval.n |
⊢ 𝐼 = ( Iso ‘ 𝐶 ) |
7 |
|
inviso1.1 |
⊢ ( 𝜑 → 𝐹 ( 𝑋 𝑁 𝑌 ) 𝐺 ) |
8 |
1 2 3 4 5
|
invfun |
⊢ ( 𝜑 → Fun ( 𝑋 𝑁 𝑌 ) ) |
9 |
|
funrel |
⊢ ( Fun ( 𝑋 𝑁 𝑌 ) → Rel ( 𝑋 𝑁 𝑌 ) ) |
10 |
8 9
|
syl |
⊢ ( 𝜑 → Rel ( 𝑋 𝑁 𝑌 ) ) |
11 |
|
releldm |
⊢ ( ( Rel ( 𝑋 𝑁 𝑌 ) ∧ 𝐹 ( 𝑋 𝑁 𝑌 ) 𝐺 ) → 𝐹 ∈ dom ( 𝑋 𝑁 𝑌 ) ) |
12 |
10 7 11
|
syl2anc |
⊢ ( 𝜑 → 𝐹 ∈ dom ( 𝑋 𝑁 𝑌 ) ) |
13 |
1 2 3 4 5 6
|
isoval |
⊢ ( 𝜑 → ( 𝑋 𝐼 𝑌 ) = dom ( 𝑋 𝑁 𝑌 ) ) |
14 |
12 13
|
eleqtrrd |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) ) |