Step |
Hyp |
Ref |
Expression |
1 |
|
odcl.1 |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
2 |
|
odcl.2 |
⊢ 𝑂 = ( od ‘ 𝐺 ) |
3 |
|
c0ex |
⊢ 0 ∈ V |
4 |
|
ltso |
⊢ < Or ℝ |
5 |
4
|
infex |
⊢ inf ( 𝑤 , ℝ , < ) ∈ V |
6 |
3 5
|
ifex |
⊢ if ( 𝑤 = ∅ , 0 , inf ( 𝑤 , ℝ , < ) ) ∈ V |
7 |
6
|
csbex |
⊢ ⦋ { 𝑧 ∈ ℕ ∣ ( 𝑧 ( .g ‘ 𝐺 ) 𝑦 ) = ( 0g ‘ 𝐺 ) } / 𝑤 ⦌ if ( 𝑤 = ∅ , 0 , inf ( 𝑤 , ℝ , < ) ) ∈ V |
8 |
|
eqid |
⊢ ( .g ‘ 𝐺 ) = ( .g ‘ 𝐺 ) |
9 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
10 |
1 8 9 2
|
odfval |
⊢ 𝑂 = ( 𝑦 ∈ 𝑋 ↦ ⦋ { 𝑧 ∈ ℕ ∣ ( 𝑧 ( .g ‘ 𝐺 ) 𝑦 ) = ( 0g ‘ 𝐺 ) } / 𝑤 ⦌ if ( 𝑤 = ∅ , 0 , inf ( 𝑤 , ℝ , < ) ) ) |
11 |
7 10
|
fnmpti |
⊢ 𝑂 Fn 𝑋 |
12 |
1 2
|
odcl |
⊢ ( 𝑥 ∈ 𝑋 → ( 𝑂 ‘ 𝑥 ) ∈ ℕ0 ) |
13 |
12
|
rgen |
⊢ ∀ 𝑥 ∈ 𝑋 ( 𝑂 ‘ 𝑥 ) ∈ ℕ0 |
14 |
|
ffnfv |
⊢ ( 𝑂 : 𝑋 ⟶ ℕ0 ↔ ( 𝑂 Fn 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ( 𝑂 ‘ 𝑥 ) ∈ ℕ0 ) ) |
15 |
11 13 14
|
mpbir2an |
⊢ 𝑂 : 𝑋 ⟶ ℕ0 |