Step |
Hyp |
Ref |
Expression |
1 |
|
cardf2 |
⊢ card : { 𝑥 ∣ ∃ 𝑦 ∈ On 𝑦 ≈ 𝑥 } ⟶ On |
2 |
|
ffun |
⊢ ( card : { 𝑥 ∣ ∃ 𝑦 ∈ On 𝑦 ≈ 𝑥 } ⟶ On → Fun card ) |
3 |
2
|
funfnd |
⊢ ( card : { 𝑥 ∣ ∃ 𝑦 ∈ On 𝑦 ≈ 𝑥 } ⟶ On → card Fn dom card ) |
4 |
1 3
|
mp1i |
⊢ ( ( 𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card ) → card Fn dom card ) |
5 |
|
fvex |
⊢ ( card ‘ 𝑦 ) ∈ V |
6 |
5
|
epeli |
⊢ ( ( card ‘ 𝑥 ) E ( card ‘ 𝑦 ) ↔ ( card ‘ 𝑥 ) ∈ ( card ‘ 𝑦 ) ) |
7 |
|
cardsdom2 |
⊢ ( ( 𝑥 ∈ dom card ∧ 𝑦 ∈ dom card ) → ( ( card ‘ 𝑥 ) ∈ ( card ‘ 𝑦 ) ↔ 𝑥 ≺ 𝑦 ) ) |
8 |
6 7
|
bitr2id |
⊢ ( ( 𝑥 ∈ dom card ∧ 𝑦 ∈ dom card ) → ( 𝑥 ≺ 𝑦 ↔ ( card ‘ 𝑥 ) E ( card ‘ 𝑦 ) ) ) |
9 |
8
|
rgen2 |
⊢ ∀ 𝑥 ∈ dom card ∀ 𝑦 ∈ dom card ( 𝑥 ≺ 𝑦 ↔ ( card ‘ 𝑥 ) E ( card ‘ 𝑦 ) ) |
10 |
9
|
a1i |
⊢ ( ( 𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card ) → ∀ 𝑥 ∈ dom card ∀ 𝑦 ∈ dom card ( 𝑥 ≺ 𝑦 ↔ ( card ‘ 𝑥 ) E ( card ‘ 𝑦 ) ) ) |
11 |
|
simpl |
⊢ ( ( 𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card ) → 𝐴 ⊆ dom card ) |
12 |
|
simpr |
⊢ ( ( 𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card ) → 𝐵 ∈ dom card ) |
13 |
4 10 11 12
|
fnrelpredd |
⊢ ( ( 𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card ) → Pred ( E , ( card “ 𝐴 ) , ( card ‘ 𝐵 ) ) = ( card “ Pred ( ≺ , 𝐴 , 𝐵 ) ) ) |