Theorem mapssfset

Description: The value of the set exponentiation ( B ^m A ) is a subset of the class of functions from A to B . (Contributed by AV, 10-Aug-2024)

	Ref	Expression
Assertion	mapssfset	⊢ ( 𝐵 ↑m 𝐴 ) ⊆ { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐵 }

Proof

Step	Hyp	Ref	Expression
1		mapfset	⊢ ( 𝐵 ∈ V → { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐵 } = ( 𝐵 ↑m 𝐴 ) )
2		eqimss2	⊢ ( { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐵 } = ( 𝐵 ↑m 𝐴 ) → ( 𝐵 ↑m 𝐴 ) ⊆ { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐵 } )
3	1 2	syl	⊢ ( 𝐵 ∈ V → ( 𝐵 ↑m 𝐴 ) ⊆ { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐵 } )
4		reldmmap	⊢ Rel dom ↑m
5	4	ovprc1	⊢ ( ¬ 𝐵 ∈ V → ( 𝐵 ↑m 𝐴 ) = ∅ )
6		0ss	⊢ ∅ ⊆ { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐵 }
7	5 6	eqsstrdi	⊢ ( ¬ 𝐵 ∈ V → ( 𝐵 ↑m 𝐴 ) ⊆ { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐵 } )
8	3 7	pm2.61i	⊢ ( 𝐵 ↑m 𝐴 ) ⊆ { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐵 }