Metamath Proof Explorer


Theorem sibff

Description: A simple function is a function. (Contributed by Thierry Arnoux, 19-Feb-2018)

Ref Expression
Hypotheses sitgval.b B = Base W
sitgval.j J = TopOpen W
sitgval.s S = 𝛔 J
sitgval.0 0 ˙ = 0 W
sitgval.x · ˙ = W
sitgval.h H = ℝHom Scalar W
sitgval.1 φ W V
sitgval.2 φ M ran measures
sibfmbl.1 φ F M W
Assertion sibff φ F : dom M J

Proof

Step Hyp Ref Expression
1 sitgval.b B = Base W
2 sitgval.j J = TopOpen W
3 sitgval.s S = 𝛔 J
4 sitgval.0 0 ˙ = 0 W
5 sitgval.x · ˙ = W
6 sitgval.h H = ℝHom Scalar W
7 sitgval.1 φ W V
8 sitgval.2 φ M ran measures
9 sibfmbl.1 φ F M W
10 dmmeas M ran measures dom M ran sigAlgebra
11 8 10 syl φ dom M ran sigAlgebra
12 fvexd φ TopOpen W V
13 2 12 eqeltrid φ J V
14 13 sgsiga φ 𝛔 J ran sigAlgebra
15 3 14 eqeltrid φ S ran sigAlgebra
16 1 2 3 4 5 6 7 8 9 sibfmbl φ F dom M MblFn μ S
17 11 15 16 mbfmf φ F : dom M S
18 3 unieqi S = 𝛔 J
19 unisg J V 𝛔 J = J
20 13 19 syl φ 𝛔 J = J
21 18 20 syl5eq φ S = J
22 21 feq3d φ F : dom M S F : dom M J
23 17 22 mpbid φ F : dom M J