Step |
Hyp |
Ref |
Expression |
1 |
|
oprres.v |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐺 𝑦 ) ) |
2 |
|
oprres.s |
⊢ ( 𝜑 → 𝑌 ⊆ 𝑋 ) |
3 |
|
oprres.f |
⊢ ( 𝜑 → 𝐹 : ( 𝑌 × 𝑌 ) ⟶ 𝑅 ) |
4 |
|
oprres.g |
⊢ ( 𝜑 → 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑆 ) |
5 |
1
|
3expb |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐺 𝑦 ) ) |
6 |
|
ovres |
⊢ ( ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ) → ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) = ( 𝑥 𝐺 𝑦 ) ) |
7 |
6
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) = ( 𝑥 𝐺 𝑦 ) ) |
8 |
5 7
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) ) |
9 |
8
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) ) |
10 |
|
eqid |
⊢ ( 𝑌 × 𝑌 ) = ( 𝑌 × 𝑌 ) |
11 |
9 10
|
jctil |
⊢ ( 𝜑 → ( ( 𝑌 × 𝑌 ) = ( 𝑌 × 𝑌 ) ∧ ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) ) ) |
12 |
3
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn ( 𝑌 × 𝑌 ) ) |
13 |
4
|
ffnd |
⊢ ( 𝜑 → 𝐺 Fn ( 𝑋 × 𝑋 ) ) |
14 |
|
xpss12 |
⊢ ( ( 𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑋 ) → ( 𝑌 × 𝑌 ) ⊆ ( 𝑋 × 𝑋 ) ) |
15 |
2 2 14
|
syl2anc |
⊢ ( 𝜑 → ( 𝑌 × 𝑌 ) ⊆ ( 𝑋 × 𝑋 ) ) |
16 |
|
fnssres |
⊢ ( ( 𝐺 Fn ( 𝑋 × 𝑋 ) ∧ ( 𝑌 × 𝑌 ) ⊆ ( 𝑋 × 𝑋 ) ) → ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) Fn ( 𝑌 × 𝑌 ) ) |
17 |
13 15 16
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) Fn ( 𝑌 × 𝑌 ) ) |
18 |
|
eqfnov |
⊢ ( ( 𝐹 Fn ( 𝑌 × 𝑌 ) ∧ ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) Fn ( 𝑌 × 𝑌 ) ) → ( 𝐹 = ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) ↔ ( ( 𝑌 × 𝑌 ) = ( 𝑌 × 𝑌 ) ∧ ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) ) ) ) |
19 |
12 17 18
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 = ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) ↔ ( ( 𝑌 × 𝑌 ) = ( 𝑌 × 𝑌 ) ∧ ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) ) ) ) |
20 |
11 19
|
mpbird |
⊢ ( 𝜑 → 𝐹 = ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) ) |