Step |
Hyp |
Ref |
Expression |
1 |
|
1on |
⊢ 1o ∈ On |
2 |
1
|
a1i |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑋 ∈ ( ℕ0 ↑m 1o ) ) → 1o ∈ On ) |
3 |
|
fvexd |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝑋 ∈ ( ℕ0 ↑m 1o ) ) ∧ 𝑎 ∈ 1o ) → ( 𝑋 ‘ ∅ ) ∈ V ) |
4 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ℕ0 ∧ 𝑋 ∈ ( ℕ0 ↑m 1o ) ) ∧ 𝑎 ∈ 1o ) → 𝐴 ∈ ℕ0 ) |
5 |
|
df1o2 |
⊢ 1o = { ∅ } |
6 |
|
nn0ex |
⊢ ℕ0 ∈ V |
7 |
|
0ex |
⊢ ∅ ∈ V |
8 |
5 6 7
|
mapsnconst |
⊢ ( 𝑋 ∈ ( ℕ0 ↑m 1o ) → 𝑋 = ( 1o × { ( 𝑋 ‘ ∅ ) } ) ) |
9 |
8
|
adantl |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑋 ∈ ( ℕ0 ↑m 1o ) ) → 𝑋 = ( 1o × { ( 𝑋 ‘ ∅ ) } ) ) |
10 |
|
fconstmpt |
⊢ ( 1o × { ( 𝑋 ‘ ∅ ) } ) = ( 𝑎 ∈ 1o ↦ ( 𝑋 ‘ ∅ ) ) |
11 |
9 10
|
eqtrdi |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑋 ∈ ( ℕ0 ↑m 1o ) ) → 𝑋 = ( 𝑎 ∈ 1o ↦ ( 𝑋 ‘ ∅ ) ) ) |
12 |
|
fconstmpt |
⊢ ( 1o × { 𝐴 } ) = ( 𝑎 ∈ 1o ↦ 𝐴 ) |
13 |
12
|
a1i |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑋 ∈ ( ℕ0 ↑m 1o ) ) → ( 1o × { 𝐴 } ) = ( 𝑎 ∈ 1o ↦ 𝐴 ) ) |
14 |
2 3 4 11 13
|
ofrfval2 |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑋 ∈ ( ℕ0 ↑m 1o ) ) → ( 𝑋 ∘r ≤ ( 1o × { 𝐴 } ) ↔ ∀ 𝑎 ∈ 1o ( 𝑋 ‘ ∅ ) ≤ 𝐴 ) ) |
15 |
|
1n0 |
⊢ 1o ≠ ∅ |
16 |
|
r19.3rzv |
⊢ ( 1o ≠ ∅ → ( ( 𝑋 ‘ ∅ ) ≤ 𝐴 ↔ ∀ 𝑎 ∈ 1o ( 𝑋 ‘ ∅ ) ≤ 𝐴 ) ) |
17 |
15 16
|
mp1i |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑋 ∈ ( ℕ0 ↑m 1o ) ) → ( ( 𝑋 ‘ ∅ ) ≤ 𝐴 ↔ ∀ 𝑎 ∈ 1o ( 𝑋 ‘ ∅ ) ≤ 𝐴 ) ) |
18 |
|
elmapi |
⊢ ( 𝑋 ∈ ( ℕ0 ↑m 1o ) → 𝑋 : 1o ⟶ ℕ0 ) |
19 |
|
0lt1o |
⊢ ∅ ∈ 1o |
20 |
|
ffvelrn |
⊢ ( ( 𝑋 : 1o ⟶ ℕ0 ∧ ∅ ∈ 1o ) → ( 𝑋 ‘ ∅ ) ∈ ℕ0 ) |
21 |
18 19 20
|
sylancl |
⊢ ( 𝑋 ∈ ( ℕ0 ↑m 1o ) → ( 𝑋 ‘ ∅ ) ∈ ℕ0 ) |
22 |
21
|
adantl |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑋 ∈ ( ℕ0 ↑m 1o ) ) → ( 𝑋 ‘ ∅ ) ∈ ℕ0 ) |
23 |
22
|
biantrurd |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑋 ∈ ( ℕ0 ↑m 1o ) ) → ( ( 𝑋 ‘ ∅ ) ≤ 𝐴 ↔ ( ( 𝑋 ‘ ∅ ) ∈ ℕ0 ∧ ( 𝑋 ‘ ∅ ) ≤ 𝐴 ) ) ) |
24 |
|
fznn0 |
⊢ ( 𝐴 ∈ ℕ0 → ( ( 𝑋 ‘ ∅ ) ∈ ( 0 ... 𝐴 ) ↔ ( ( 𝑋 ‘ ∅ ) ∈ ℕ0 ∧ ( 𝑋 ‘ ∅ ) ≤ 𝐴 ) ) ) |
25 |
24
|
adantr |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑋 ∈ ( ℕ0 ↑m 1o ) ) → ( ( 𝑋 ‘ ∅ ) ∈ ( 0 ... 𝐴 ) ↔ ( ( 𝑋 ‘ ∅ ) ∈ ℕ0 ∧ ( 𝑋 ‘ ∅ ) ≤ 𝐴 ) ) ) |
26 |
23 25
|
bitr4d |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑋 ∈ ( ℕ0 ↑m 1o ) ) → ( ( 𝑋 ‘ ∅ ) ≤ 𝐴 ↔ ( 𝑋 ‘ ∅ ) ∈ ( 0 ... 𝐴 ) ) ) |
27 |
14 17 26
|
3bitr2d |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝑋 ∈ ( ℕ0 ↑m 1o ) ) → ( 𝑋 ∘r ≤ ( 1o × { 𝐴 } ) ↔ ( 𝑋 ‘ ∅ ) ∈ ( 0 ... 𝐴 ) ) ) |