Metamath Proof Explorer


Theorem smfpreimage

Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of a closed interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021)

Ref Expression
Hypotheses smfpreimage.s φ S SAlg
smfpreimage.f φ F SMblFn S
smfpreimage.d D = dom F
smfpreimage.a φ A
Assertion smfpreimage φ x D | A F x S 𝑡 D

Proof

Step Hyp Ref Expression
1 smfpreimage.s φ S SAlg
2 smfpreimage.f φ F SMblFn S
3 smfpreimage.d D = dom F
4 smfpreimage.a φ A
5 1 3 issmfge φ F SMblFn S D S F : D a x D | a F x S 𝑡 D
6 2 5 mpbid φ D S F : D a x D | a F x S 𝑡 D
7 6 simp3d φ a x D | a F x S 𝑡 D
8 breq1 a = A a F x A F x
9 8 rabbidv a = A x D | a F x = x D | A F x
10 9 eleq1d a = A x D | a F x S 𝑡 D x D | A F x S 𝑡 D
11 10 rspcva A a x D | a F x S 𝑡 D x D | A F x S 𝑡 D
12 4 7 11 syl2anc φ x D | A F x S 𝑡 D