Step |
Hyp |
Ref |
Expression |
1 |
|
caofref.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
caofref.2 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 ) |
3 |
|
caofid0.3 |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
4 |
|
caofid0l.5 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → ( 𝐵 𝑅 𝑥 ) = 𝑥 ) |
5 |
|
fnconstg |
⊢ ( 𝐵 ∈ 𝑊 → ( 𝐴 × { 𝐵 } ) Fn 𝐴 ) |
6 |
3 5
|
syl |
⊢ ( 𝜑 → ( 𝐴 × { 𝐵 } ) Fn 𝐴 ) |
7 |
2
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
8 |
|
fvconst2g |
⊢ ( ( 𝐵 ∈ 𝑊 ∧ 𝑤 ∈ 𝐴 ) → ( ( 𝐴 × { 𝐵 } ) ‘ 𝑤 ) = 𝐵 ) |
9 |
3 8
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( ( 𝐴 × { 𝐵 } ) ‘ 𝑤 ) = 𝐵 ) |
10 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑤 ) ) |
11 |
4
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ( 𝐵 𝑅 𝑥 ) = 𝑥 ) |
12 |
2
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑤 ) ∈ 𝑆 ) |
13 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑤 ) → ( 𝐵 𝑅 𝑥 ) = ( 𝐵 𝑅 ( 𝐹 ‘ 𝑤 ) ) ) |
14 |
|
id |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑤 ) → 𝑥 = ( 𝐹 ‘ 𝑤 ) ) |
15 |
13 14
|
eqeq12d |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑤 ) → ( ( 𝐵 𝑅 𝑥 ) = 𝑥 ↔ ( 𝐵 𝑅 ( 𝐹 ‘ 𝑤 ) ) = ( 𝐹 ‘ 𝑤 ) ) ) |
16 |
15
|
rspccva |
⊢ ( ( ∀ 𝑥 ∈ 𝑆 ( 𝐵 𝑅 𝑥 ) = 𝑥 ∧ ( 𝐹 ‘ 𝑤 ) ∈ 𝑆 ) → ( 𝐵 𝑅 ( 𝐹 ‘ 𝑤 ) ) = ( 𝐹 ‘ 𝑤 ) ) |
17 |
11 12 16
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐵 𝑅 ( 𝐹 ‘ 𝑤 ) ) = ( 𝐹 ‘ 𝑤 ) ) |
18 |
1 6 7 7 9 10 17
|
offveq |
⊢ ( 𝜑 → ( ( 𝐴 × { 𝐵 } ) ∘f 𝑅 𝐹 ) = 𝐹 ) |