Theorem fsovf1od

Description: The value of ( A O B ) is a bijection, where O is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets. (Contributed by RP, 27-Apr-2021)

	Ref	Expression
Hypotheses	fsovd.fs	⊢ 𝑂 = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑓 ∈ ( 𝒫 𝑏 ↑m 𝑎 ) ↦ ( 𝑦 ∈ 𝑏 ↦ { 𝑥 ∈ 𝑎 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) ) )
	fsovd.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	fsovd.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
	fsovfvd.g	⊢ 𝐺 = ( 𝐴 𝑂 𝐵 )
Assertion	fsovf1od	⊢ ( 𝜑 → 𝐺 : ( 𝒫 𝐵 ↑m 𝐴 ) –1-1-onto→ ( 𝒫 𝐴 ↑m 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fsovd.fs	⊢ 𝑂 = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑓 ∈ ( 𝒫 𝑏 ↑m 𝑎 ) ↦ ( 𝑦 ∈ 𝑏 ↦ { 𝑥 ∈ 𝑎 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) ) )
2		fsovd.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
3		fsovd.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
4		fsovfvd.g	⊢ 𝐺 = ( 𝐴 𝑂 𝐵 )
5	1 2 3 4	fsovfd	⊢ ( 𝜑 → 𝐺 : ( 𝒫 𝐵 ↑m 𝐴 ) ⟶ ( 𝒫 𝐴 ↑m 𝐵 ) )
6	5	ffnd	⊢ ( 𝜑 → 𝐺 Fn ( 𝒫 𝐵 ↑m 𝐴 ) )
7		eqid	⊢ ( 𝐵 𝑂 𝐴 ) = ( 𝐵 𝑂 𝐴 )
8	1 3 2 7	fsovfd	⊢ ( 𝜑 → ( 𝐵 𝑂 𝐴 ) : ( 𝒫 𝐴 ↑m 𝐵 ) ⟶ ( 𝒫 𝐵 ↑m 𝐴 ) )
9	8	ffnd	⊢ ( 𝜑 → ( 𝐵 𝑂 𝐴 ) Fn ( 𝒫 𝐴 ↑m 𝐵 ) )
10	1 2 3 4 7	fsovcnvd	⊢ ( 𝜑 → ◡ 𝐺 = ( 𝐵 𝑂 𝐴 ) )
11	10	fneq1d	⊢ ( 𝜑 → ( ◡ 𝐺 Fn ( 𝒫 𝐴 ↑m 𝐵 ) ↔ ( 𝐵 𝑂 𝐴 ) Fn ( 𝒫 𝐴 ↑m 𝐵 ) ) )
12	9 11	mpbird	⊢ ( 𝜑 → ◡ 𝐺 Fn ( 𝒫 𝐴 ↑m 𝐵 ) )
13		dff1o4	⊢ ( 𝐺 : ( 𝒫 𝐵 ↑m 𝐴 ) –1-1-onto→ ( 𝒫 𝐴 ↑m 𝐵 ) ↔ ( 𝐺 Fn ( 𝒫 𝐵 ↑m 𝐴 ) ∧ ◡ 𝐺 Fn ( 𝒫 𝐴 ↑m 𝐵 ) ) )
14	6 12 13	sylanbrc	⊢ ( 𝜑 → 𝐺 : ( 𝒫 𝐵 ↑m 𝐴 ) –1-1-onto→ ( 𝒫 𝐴 ↑m 𝐵 ) )