Theorem smffmpt

Description: A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	smffmpt.x	⊢ Ⅎ 𝑥 𝜑
	smffmpt.s	⊢ ( 𝜑 → 𝑆 ∈ SAlg )
	smffmpt.b	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
	smffmpt.m	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) )
Assertion	smffmpt	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℝ )

Proof

Step	Hyp	Ref	Expression
1		smffmpt.x	⊢ Ⅎ 𝑥 𝜑
2		smffmpt.s	⊢ ( 𝜑 → 𝑆 ∈ SAlg )
3		smffmpt.b	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
4		smffmpt.m	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) )
5		eqid	⊢ dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
6	2 4 5	smff	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⟶ ℝ )
7		eqid	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
8	1 7 3	dmmptdf	⊢ ( 𝜑 → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 )
9	8	eqcomd	⊢ ( 𝜑 → 𝐴 = dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
10	9	feq2d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℝ ↔ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⟶ ℝ ) )
11	6 10	mpbird	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℝ )