Theorem enrelmapr

Description: The set of all possible relations between two sets is equinumerous to the set of all mappings from one set to the powerset of the other. (Contributed by RP, 27-Apr-2021)

	Ref	Expression
Assertion	enrelmapr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → 𝒫 ( 𝐴 × 𝐵 ) ≈ ( 𝒫 𝐴 ↑m 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		xpcomeng	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 × 𝐵 ) ≈ ( 𝐵 × 𝐴 ) )
2		pwen	⊢ ( ( 𝐴 × 𝐵 ) ≈ ( 𝐵 × 𝐴 ) → 𝒫 ( 𝐴 × 𝐵 ) ≈ 𝒫 ( 𝐵 × 𝐴 ) )
3	1 2	syl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → 𝒫 ( 𝐴 × 𝐵 ) ≈ 𝒫 ( 𝐵 × 𝐴 ) )
4		enrelmap	⊢ ( ( 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ) → 𝒫 ( 𝐵 × 𝐴 ) ≈ ( 𝒫 𝐴 ↑m 𝐵 ) )
5	4	ancoms	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → 𝒫 ( 𝐵 × 𝐴 ) ≈ ( 𝒫 𝐴 ↑m 𝐵 ) )
6		entr	⊢ ( ( 𝒫 ( 𝐴 × 𝐵 ) ≈ 𝒫 ( 𝐵 × 𝐴 ) ∧ 𝒫 ( 𝐵 × 𝐴 ) ≈ ( 𝒫 𝐴 ↑m 𝐵 ) ) → 𝒫 ( 𝐴 × 𝐵 ) ≈ ( 𝒫 𝐴 ↑m 𝐵 ) )
7	3 5 6	syl2anc	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → 𝒫 ( 𝐴 × 𝐵 ) ≈ ( 𝒫 𝐴 ↑m 𝐵 ) )