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	|- F/ x ph
	smffmpt.s	|- ( ph -> S e. SAlg )
	smffmpt.b	|- ( ( ph /\ x e. A ) -> B e. V )
	smffmpt.m	|- ( ph -> ( x e. A |-> B ) e. ( SMblFn ` S ) )
Assertion	smffmpt	|- ( ph -> ( x e. A |-> B ) : A --> RR )

Proof

Step	Hyp	Ref	Expression
1		smffmpt.x	|- F/ x ph
2		smffmpt.s	|- ( ph -> S e. SAlg )
3		smffmpt.b	|- ( ( ph /\ x e. A ) -> B e. V )
4		smffmpt.m	|- ( ph -> ( x e. A |-> B ) e. ( SMblFn ` S ) )
5		eqid	|- dom ( x e. A |-> B ) = dom ( x e. A |-> B )
6	2 4 5	smff	|- ( ph -> ( x e. A |-> B ) : dom ( x e. A |-> B ) --> RR )
7		eqid	|- ( x e. A |-> B ) = ( x e. A |-> B )
8	1 7 3	dmmptdf	|- ( ph -> dom ( x e. A |-> B ) = A )
9	8	eqcomd	|- ( ph -> A = dom ( x e. A |-> B ) )
10	9	feq2d	|- ( ph -> ( ( x e. A |-> B ) : A --> RR <-> ( x e. A |-> B ) : dom ( x e. A |-> B ) --> RR ) )
11	6 10	mpbird	|- ( ph -> ( x e. A |-> B ) : A --> RR )