Step |
Hyp |
Ref |
Expression |
1 |
|
caofref.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
caofref.2 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 ) |
3 |
|
caofid0.3 |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
4 |
|
caofid1.4 |
⊢ ( 𝜑 → 𝐶 ∈ 𝑋 ) |
5 |
|
caofid2.5 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → ( 𝐵 𝑅 𝑥 ) = 𝐶 ) |
6 |
|
fnconstg |
⊢ ( 𝐵 ∈ 𝑊 → ( 𝐴 × { 𝐵 } ) Fn 𝐴 ) |
7 |
3 6
|
syl |
⊢ ( 𝜑 → ( 𝐴 × { 𝐵 } ) Fn 𝐴 ) |
8 |
2
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
9 |
|
fnconstg |
⊢ ( 𝐶 ∈ 𝑋 → ( 𝐴 × { 𝐶 } ) Fn 𝐴 ) |
10 |
4 9
|
syl |
⊢ ( 𝜑 → ( 𝐴 × { 𝐶 } ) Fn 𝐴 ) |
11 |
|
fvconst2g |
⊢ ( ( 𝐵 ∈ 𝑊 ∧ 𝑤 ∈ 𝐴 ) → ( ( 𝐴 × { 𝐵 } ) ‘ 𝑤 ) = 𝐵 ) |
12 |
3 11
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( ( 𝐴 × { 𝐵 } ) ‘ 𝑤 ) = 𝐵 ) |
13 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑤 ) ) |
14 |
5
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ( 𝐵 𝑅 𝑥 ) = 𝐶 ) |
15 |
2
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑤 ) ∈ 𝑆 ) |
16 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑤 ) → ( 𝐵 𝑅 𝑥 ) = ( 𝐵 𝑅 ( 𝐹 ‘ 𝑤 ) ) ) |
17 |
16
|
eqeq1d |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑤 ) → ( ( 𝐵 𝑅 𝑥 ) = 𝐶 ↔ ( 𝐵 𝑅 ( 𝐹 ‘ 𝑤 ) ) = 𝐶 ) ) |
18 |
17
|
rspccva |
⊢ ( ( ∀ 𝑥 ∈ 𝑆 ( 𝐵 𝑅 𝑥 ) = 𝐶 ∧ ( 𝐹 ‘ 𝑤 ) ∈ 𝑆 ) → ( 𝐵 𝑅 ( 𝐹 ‘ 𝑤 ) ) = 𝐶 ) |
19 |
14 15 18
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐵 𝑅 ( 𝐹 ‘ 𝑤 ) ) = 𝐶 ) |
20 |
|
fvconst2g |
⊢ ( ( 𝐶 ∈ 𝑋 ∧ 𝑤 ∈ 𝐴 ) → ( ( 𝐴 × { 𝐶 } ) ‘ 𝑤 ) = 𝐶 ) |
21 |
4 20
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( ( 𝐴 × { 𝐶 } ) ‘ 𝑤 ) = 𝐶 ) |
22 |
19 21
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝐵 𝑅 ( 𝐹 ‘ 𝑤 ) ) = ( ( 𝐴 × { 𝐶 } ) ‘ 𝑤 ) ) |
23 |
1 7 8 10 12 13 22
|
offveq |
⊢ ( 𝜑 → ( ( 𝐴 × { 𝐵 } ) ∘f 𝑅 𝐹 ) = ( 𝐴 × { 𝐶 } ) ) |