Step |
Hyp |
Ref |
Expression |
1 |
|
caofref.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
caofref.2 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 ) |
3 |
|
caofcom.3 |
⊢ ( 𝜑 → 𝐺 : 𝐴 ⟶ 𝑆 ) |
4 |
|
caofrss.4 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝑅 𝑦 → 𝑥 𝑇 𝑦 ) ) |
5 |
2
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑤 ) ∈ 𝑆 ) |
6 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑤 ) ∈ 𝑆 ) |
7 |
4
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝑅 𝑦 → 𝑥 𝑇 𝑦 ) ) |
8 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝑅 𝑦 → 𝑥 𝑇 𝑦 ) ) |
9 |
|
breq1 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑤 ) → ( 𝑥 𝑅 𝑦 ↔ ( 𝐹 ‘ 𝑤 ) 𝑅 𝑦 ) ) |
10 |
|
breq1 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑤 ) → ( 𝑥 𝑇 𝑦 ↔ ( 𝐹 ‘ 𝑤 ) 𝑇 𝑦 ) ) |
11 |
9 10
|
imbi12d |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑤 ) → ( ( 𝑥 𝑅 𝑦 → 𝑥 𝑇 𝑦 ) ↔ ( ( 𝐹 ‘ 𝑤 ) 𝑅 𝑦 → ( 𝐹 ‘ 𝑤 ) 𝑇 𝑦 ) ) ) |
12 |
|
breq2 |
⊢ ( 𝑦 = ( 𝐺 ‘ 𝑤 ) → ( ( 𝐹 ‘ 𝑤 ) 𝑅 𝑦 ↔ ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ) ) |
13 |
|
breq2 |
⊢ ( 𝑦 = ( 𝐺 ‘ 𝑤 ) → ( ( 𝐹 ‘ 𝑤 ) 𝑇 𝑦 ↔ ( 𝐹 ‘ 𝑤 ) 𝑇 ( 𝐺 ‘ 𝑤 ) ) ) |
14 |
12 13
|
imbi12d |
⊢ ( 𝑦 = ( 𝐺 ‘ 𝑤 ) → ( ( ( 𝐹 ‘ 𝑤 ) 𝑅 𝑦 → ( 𝐹 ‘ 𝑤 ) 𝑇 𝑦 ) ↔ ( ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) → ( 𝐹 ‘ 𝑤 ) 𝑇 ( 𝐺 ‘ 𝑤 ) ) ) ) |
15 |
11 14
|
rspc2va |
⊢ ( ( ( ( 𝐹 ‘ 𝑤 ) ∈ 𝑆 ∧ ( 𝐺 ‘ 𝑤 ) ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝑅 𝑦 → 𝑥 𝑇 𝑦 ) ) → ( ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) → ( 𝐹 ‘ 𝑤 ) 𝑇 ( 𝐺 ‘ 𝑤 ) ) ) |
16 |
5 6 8 15
|
syl21anc |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) → ( 𝐹 ‘ 𝑤 ) 𝑇 ( 𝐺 ‘ 𝑤 ) ) ) |
17 |
16
|
ralimdva |
⊢ ( 𝜑 → ( ∀ 𝑤 ∈ 𝐴 ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) → ∀ 𝑤 ∈ 𝐴 ( 𝐹 ‘ 𝑤 ) 𝑇 ( 𝐺 ‘ 𝑤 ) ) ) |
18 |
2
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
19 |
3
|
ffnd |
⊢ ( 𝜑 → 𝐺 Fn 𝐴 ) |
20 |
|
inidm |
⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴 |
21 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑤 ) ) |
22 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑤 ) = ( 𝐺 ‘ 𝑤 ) ) |
23 |
18 19 1 1 20 21 22
|
ofrfval |
⊢ ( 𝜑 → ( 𝐹 ∘r 𝑅 𝐺 ↔ ∀ 𝑤 ∈ 𝐴 ( 𝐹 ‘ 𝑤 ) 𝑅 ( 𝐺 ‘ 𝑤 ) ) ) |
24 |
18 19 1 1 20 21 22
|
ofrfval |
⊢ ( 𝜑 → ( 𝐹 ∘r 𝑇 𝐺 ↔ ∀ 𝑤 ∈ 𝐴 ( 𝐹 ‘ 𝑤 ) 𝑇 ( 𝐺 ‘ 𝑤 ) ) ) |
25 |
17 23 24
|
3imtr4d |
⊢ ( 𝜑 → ( 𝐹 ∘r 𝑅 𝐺 → 𝐹 ∘r 𝑇 𝐺 ) ) |