Theorem sloteq

Description: Equality theorem for the Slot construction. The converse holds if A (or B ) is a set. (Contributed by BJ, 27-Dec-2021)

	Ref	Expression
Assertion	sloteq	⊢ ( 𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( 𝐴 = 𝐵 → ( 𝑓 ‘ 𝐴 ) = ( 𝑓 ‘ 𝐵 ) )
2	1	mpteq2dv	⊢ ( 𝐴 = 𝐵 → ( 𝑓 ∈ V ↦ ( 𝑓 ‘ 𝐴 ) ) = ( 𝑓 ∈ V ↦ ( 𝑓 ‘ 𝐵 ) ) )
3		df-slot	⊢ Slot 𝐴 = ( 𝑓 ∈ V ↦ ( 𝑓 ‘ 𝐴 ) )
4		df-slot	⊢ Slot 𝐵 = ( 𝑓 ∈ V ↦ ( 𝑓 ‘ 𝐵 ) )
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵 )