Metamath Proof Explorer


Theorem cardpred

Description: The cardinality function preserves predecessors. (Contributed by BTernaryTau, 18-Jul-2024)

Ref Expression
Assertion cardpred ( ( 𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card ) → Pred ( E , ( card “ 𝐴 ) , ( card ‘ 𝐵 ) ) = ( card “ Pred ( ≺ , 𝐴 , 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 cardf2 card : { 𝑥 ∣ ∃ 𝑦 ∈ On 𝑦𝑥 } ⟶ On
2 ffun ( card : { 𝑥 ∣ ∃ 𝑦 ∈ On 𝑦𝑥 } ⟶ On → Fun card )
3 2 funfnd ( card : { 𝑥 ∣ ∃ 𝑦 ∈ On 𝑦𝑥 } ⟶ 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 ) ↔ 𝑥𝑦 ) )
8 6 7 bitr2id ( ( 𝑥 ∈ dom card ∧ 𝑦 ∈ dom card ) → ( 𝑥𝑦 ↔ ( card ‘ 𝑥 ) E ( card ‘ 𝑦 ) ) )
9 8 rgen2 𝑥 ∈ dom card ∀ 𝑦 ∈ dom card ( 𝑥𝑦 ↔ ( card ‘ 𝑥 ) E ( card ‘ 𝑦 ) )
10 9 a1i ( ( 𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card ) → ∀ 𝑥 ∈ dom card ∀ 𝑦 ∈ dom card ( 𝑥𝑦 ↔ ( 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 ) ) )