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	$$⊢Ⅎx\phi$$
	smffmpt.s	$$⊢\phi \to S\in \mathrm{SAlg}$$
	smffmpt.b	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
	smffmpt.m	$$⊢\phi \to \left(x\in A⟼B\right)\in \mathrm{SMblFn}\left(S\right)$$
Assertion	smffmpt	$$⊢\phi \to \left(x\in A⟼B\right):A⟶ℝ$$

Proof

Step	Hyp	Ref	Expression
1		smffmpt.x	$$⊢Ⅎx\phi$$
2		smffmpt.s	$$⊢\phi \to S\in \mathrm{SAlg}$$
3		smffmpt.b	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
4		smffmpt.m	$$⊢\phi \to \left(x\in A⟼B\right)\in \mathrm{SMblFn}\left(S\right)$$
5		eqid	$$⊢\mathrm{dom}\left(x\in A⟼B\right)=\mathrm{dom}\left(x\in A⟼B\right)$$
6	2 4 5	smff	$$⊢\phi \to \left(x\in A⟼B\right):\mathrm{dom}\left(x\in A⟼B\right)⟶ℝ$$
7		eqid	$$⊢\left(x\in A⟼B\right)=\left(x\in A⟼B\right)$$
8	1 7 3	dmmptdf	$$⊢\phi \to \mathrm{dom}\left(x\in A⟼B\right)=A$$
9	8	eqcomd	$$⊢\phi \to A=\mathrm{dom}\left(x\in A⟼B\right)$$
10	9	feq2d	$$⊢\phi \to \left(\left(x\in A⟼B\right):A⟶ℝ\leftrightarrow \left(x\in A⟼B\right):\mathrm{dom}\left(x\in A⟼B\right)⟶ℝ\right)$$
11	6 10	mpbird	$$⊢\phi \to \left(x\in A⟼B\right):A⟶ℝ$$