Step |
Hyp |
Ref |
Expression |
1 |
|
uniimadom.1 |
⊢ 𝐴 ∈ V |
2 |
|
uniimadom.2 |
⊢ 𝐵 ∈ V |
3 |
1
|
funimaex |
⊢ ( Fun 𝐹 → ( 𝐹 “ 𝐴 ) ∈ V ) |
4 |
3
|
adantr |
⊢ ( ( Fun 𝐹 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ≼ 𝐵 ) → ( 𝐹 “ 𝐴 ) ∈ V ) |
5 |
|
fvelima |
⊢ ( ( Fun 𝐹 ∧ 𝑦 ∈ ( 𝐹 “ 𝐴 ) ) → ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) |
6 |
5
|
ex |
⊢ ( Fun 𝐹 → ( 𝑦 ∈ ( 𝐹 “ 𝐴 ) → ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) |
7 |
|
breq1 |
⊢ ( ( 𝐹 ‘ 𝑥 ) = 𝑦 → ( ( 𝐹 ‘ 𝑥 ) ≼ 𝐵 ↔ 𝑦 ≼ 𝐵 ) ) |
8 |
7
|
biimpd |
⊢ ( ( 𝐹 ‘ 𝑥 ) = 𝑦 → ( ( 𝐹 ‘ 𝑥 ) ≼ 𝐵 → 𝑦 ≼ 𝐵 ) ) |
9 |
8
|
reximi |
⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 → ∃ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) ≼ 𝐵 → 𝑦 ≼ 𝐵 ) ) |
10 |
|
r19.36v |
⊢ ( ∃ 𝑥 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) ≼ 𝐵 → 𝑦 ≼ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ≼ 𝐵 → 𝑦 ≼ 𝐵 ) ) |
11 |
9 10
|
syl |
⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 → ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ≼ 𝐵 → 𝑦 ≼ 𝐵 ) ) |
12 |
6 11
|
syl6 |
⊢ ( Fun 𝐹 → ( 𝑦 ∈ ( 𝐹 “ 𝐴 ) → ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ≼ 𝐵 → 𝑦 ≼ 𝐵 ) ) ) |
13 |
12
|
com23 |
⊢ ( Fun 𝐹 → ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ≼ 𝐵 → ( 𝑦 ∈ ( 𝐹 “ 𝐴 ) → 𝑦 ≼ 𝐵 ) ) ) |
14 |
13
|
imp |
⊢ ( ( Fun 𝐹 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ≼ 𝐵 ) → ( 𝑦 ∈ ( 𝐹 “ 𝐴 ) → 𝑦 ≼ 𝐵 ) ) |
15 |
14
|
ralrimiv |
⊢ ( ( Fun 𝐹 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ≼ 𝐵 ) → ∀ 𝑦 ∈ ( 𝐹 “ 𝐴 ) 𝑦 ≼ 𝐵 ) |
16 |
|
unidom |
⊢ ( ( ( 𝐹 “ 𝐴 ) ∈ V ∧ ∀ 𝑦 ∈ ( 𝐹 “ 𝐴 ) 𝑦 ≼ 𝐵 ) → ∪ ( 𝐹 “ 𝐴 ) ≼ ( ( 𝐹 “ 𝐴 ) × 𝐵 ) ) |
17 |
4 15 16
|
syl2anc |
⊢ ( ( Fun 𝐹 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ≼ 𝐵 ) → ∪ ( 𝐹 “ 𝐴 ) ≼ ( ( 𝐹 “ 𝐴 ) × 𝐵 ) ) |
18 |
|
imadomg |
⊢ ( 𝐴 ∈ V → ( Fun 𝐹 → ( 𝐹 “ 𝐴 ) ≼ 𝐴 ) ) |
19 |
1 18
|
ax-mp |
⊢ ( Fun 𝐹 → ( 𝐹 “ 𝐴 ) ≼ 𝐴 ) |
20 |
2
|
xpdom1 |
⊢ ( ( 𝐹 “ 𝐴 ) ≼ 𝐴 → ( ( 𝐹 “ 𝐴 ) × 𝐵 ) ≼ ( 𝐴 × 𝐵 ) ) |
21 |
19 20
|
syl |
⊢ ( Fun 𝐹 → ( ( 𝐹 “ 𝐴 ) × 𝐵 ) ≼ ( 𝐴 × 𝐵 ) ) |
22 |
21
|
adantr |
⊢ ( ( Fun 𝐹 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ≼ 𝐵 ) → ( ( 𝐹 “ 𝐴 ) × 𝐵 ) ≼ ( 𝐴 × 𝐵 ) ) |
23 |
|
domtr |
⊢ ( ( ∪ ( 𝐹 “ 𝐴 ) ≼ ( ( 𝐹 “ 𝐴 ) × 𝐵 ) ∧ ( ( 𝐹 “ 𝐴 ) × 𝐵 ) ≼ ( 𝐴 × 𝐵 ) ) → ∪ ( 𝐹 “ 𝐴 ) ≼ ( 𝐴 × 𝐵 ) ) |
24 |
17 22 23
|
syl2anc |
⊢ ( ( Fun 𝐹 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ≼ 𝐵 ) → ∪ ( 𝐹 “ 𝐴 ) ≼ ( 𝐴 × 𝐵 ) ) |