Step |
Hyp |
Ref |
Expression |
1 |
|
ismntd.1 |
⊢ 𝐴 = ( Base ‘ 𝑉 ) |
2 |
|
ismntd.2 |
⊢ 𝐵 = ( Base ‘ 𝑊 ) |
3 |
|
ismntd.3 |
⊢ ≤ = ( le ‘ 𝑉 ) |
4 |
|
ismntd.4 |
⊢ ≲ = ( le ‘ 𝑊 ) |
5 |
|
ismntd.5 |
⊢ ( 𝜑 → 𝑉 ∈ 𝐶 ) |
6 |
|
ismntd.6 |
⊢ ( 𝜑 → 𝑊 ∈ 𝐷 ) |
7 |
|
ismntd.7 |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝑉 Monot 𝑊 ) ) |
8 |
|
ismntd.8 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐴 ) |
9 |
|
ismntd.9 |
⊢ ( 𝜑 → 𝑌 ∈ 𝐴 ) |
10 |
|
ismntd.10 |
⊢ ( 𝜑 → 𝑋 ≤ 𝑌 ) |
11 |
1 2 3 4
|
ismnt |
⊢ ( ( 𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ) → ( 𝐹 ∈ ( 𝑉 Monot 𝑊 ) ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≲ ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
12 |
11
|
biimp3a |
⊢ ( ( 𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝐹 ∈ ( 𝑉 Monot 𝑊 ) ) → ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≲ ( 𝐹 ‘ 𝑦 ) ) ) ) |
13 |
12
|
simprd |
⊢ ( ( 𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝐹 ∈ ( 𝑉 Monot 𝑊 ) ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≲ ( 𝐹 ‘ 𝑦 ) ) ) |
14 |
5 6 7 13
|
syl3anc |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≲ ( 𝐹 ‘ 𝑦 ) ) ) |
15 |
|
breq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦 ) ) |
16 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑋 ) ) |
17 |
16
|
breq1d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝐹 ‘ 𝑥 ) ≲ ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑋 ) ≲ ( 𝐹 ‘ 𝑦 ) ) ) |
18 |
15 17
|
imbi12d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≲ ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑋 ≤ 𝑦 → ( 𝐹 ‘ 𝑋 ) ≲ ( 𝐹 ‘ 𝑦 ) ) ) ) |
19 |
|
breq2 |
⊢ ( 𝑦 = 𝑌 → ( 𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌 ) ) |
20 |
|
fveq2 |
⊢ ( 𝑦 = 𝑌 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑌 ) ) |
21 |
20
|
breq2d |
⊢ ( 𝑦 = 𝑌 → ( ( 𝐹 ‘ 𝑋 ) ≲ ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑋 ) ≲ ( 𝐹 ‘ 𝑌 ) ) ) |
22 |
19 21
|
imbi12d |
⊢ ( 𝑦 = 𝑌 → ( ( 𝑋 ≤ 𝑦 → ( 𝐹 ‘ 𝑋 ) ≲ ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑋 ≤ 𝑌 → ( 𝐹 ‘ 𝑋 ) ≲ ( 𝐹 ‘ 𝑌 ) ) ) ) |
23 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → 𝐴 = 𝐴 ) |
24 |
18 22 8 23 9
|
rspc2vd |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ≤ 𝑦 → ( 𝐹 ‘ 𝑥 ) ≲ ( 𝐹 ‘ 𝑦 ) ) → ( 𝑋 ≤ 𝑌 → ( 𝐹 ‘ 𝑋 ) ≲ ( 𝐹 ‘ 𝑌 ) ) ) ) |
25 |
14 10 24
|
mp2d |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ≲ ( 𝐹 ‘ 𝑌 ) ) |