Step |
Hyp |
Ref |
Expression |
1 |
|
caofref.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
caofref.2 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 ) |
3 |
|
caofref.3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → 𝑥 𝑅 𝑥 ) |
4 |
|
id |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑤 ) → 𝑥 = ( 𝐹 ‘ 𝑤 ) ) |
5 |
4 4
|
breq12d |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑤 ) → ( 𝑥 𝑅 𝑥 ↔ ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐹 ‘ 𝑤 ) ) ) |
6 |
3
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 𝑥 𝑅 𝑥 ) |
7 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ∀ 𝑥 ∈ 𝑆 𝑥 𝑅 𝑥 ) |
8 |
2
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑤 ) ∈ 𝑆 ) |
9 |
5 7 8
|
rspcdva |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐹 ‘ 𝑤 ) ) |
10 |
9
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑤 ∈ 𝐴 ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐹 ‘ 𝑤 ) ) |
11 |
2
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
12 |
|
inidm |
⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴 |
13 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑤 ) ) |
14 |
11 11 1 1 12 13 13
|
ofrfval |
⊢ ( 𝜑 → ( 𝐹 ∘r 𝑅 𝐹 ↔ ∀ 𝑤 ∈ 𝐴 ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐹 ‘ 𝑤 ) ) ) |
15 |
10 14
|
mpbird |
⊢ ( 𝜑 → 𝐹 ∘r 𝑅 𝐹 ) |