Theorem fsovd

Description: Value of the operator, ( A O B ) , 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, A and B . (Contributed by RP, 25-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 )
Assertion	fsovd	⊢ ( 𝜑 → ( 𝐴 𝑂 𝐵 ) = ( 𝑓 ∈ ( 𝒫 𝐵 ↑m 𝐴 ) ↦ ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) ) )

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	1	a1i	⊢ ( 𝜑 → 𝑂 = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑓 ∈ ( 𝒫 𝑏 ↑m 𝑎 ) ↦ ( 𝑦 ∈ 𝑏 ↦ { 𝑥 ∈ 𝑎 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) ) ) )
5		pweq	⊢ ( 𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵 )
6	5	adantl	⊢ ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → 𝒫 𝑏 = 𝒫 𝐵 )
7		simpl	⊢ ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → 𝑎 = 𝐴 )
8	6 7	oveq12d	⊢ ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → ( 𝒫 𝑏 ↑m 𝑎 ) = ( 𝒫 𝐵 ↑m 𝐴 ) )
9		simpr	⊢ ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → 𝑏 = 𝐵 )
10		rabeq	⊢ ( 𝑎 = 𝐴 → { 𝑥 ∈ 𝑎 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } = { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } )
11	10	adantr	⊢ ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → { 𝑥 ∈ 𝑎 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } = { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } )
12	9 11	mpteq12dv	⊢ ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → ( 𝑦 ∈ 𝑏 ↦ { 𝑥 ∈ 𝑎 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) = ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) )
13	8 12	mpteq12dv	⊢ ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → ( 𝑓 ∈ ( 𝒫 𝑏 ↑m 𝑎 ) ↦ ( 𝑦 ∈ 𝑏 ↦ { 𝑥 ∈ 𝑎 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) ) = ( 𝑓 ∈ ( 𝒫 𝐵 ↑m 𝐴 ) ↦ ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) ) )
14	13	adantl	⊢ ( ( 𝜑 ∧ ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ) → ( 𝑓 ∈ ( 𝒫 𝑏 ↑m 𝑎 ) ↦ ( 𝑦 ∈ 𝑏 ↦ { 𝑥 ∈ 𝑎 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) ) = ( 𝑓 ∈ ( 𝒫 𝐵 ↑m 𝐴 ) ↦ ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) ) )
15	2	elexd	⊢ ( 𝜑 → 𝐴 ∈ V )
16	3	elexd	⊢ ( 𝜑 → 𝐵 ∈ V )
17		ovex	⊢ ( 𝒫 𝐵 ↑m 𝐴 ) ∈ V
18	17	mptex	⊢ ( 𝑓 ∈ ( 𝒫 𝐵 ↑m 𝐴 ) ↦ ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) ) ∈ V
19	18	a1i	⊢ ( 𝜑 → ( 𝑓 ∈ ( 𝒫 𝐵 ↑m 𝐴 ) ↦ ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) ) ∈ V )
20	4 14 15 16 19	ovmpod	⊢ ( 𝜑 → ( 𝐴 𝑂 𝐵 ) = ( 𝑓 ∈ ( 𝒫 𝐵 ↑m 𝐴 ) ↦ ( 𝑦 ∈ 𝐵 ↦ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝑓 ‘ 𝑥 ) } ) ) )