Theorem mbfmbfm 34442

Description: A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.) Remove hypotheses. (Revised by SN, 13-Jan-2025.)

Hypothesis

Ref	Expression
mbfmbfm.1	⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))

Assertion

Ref	Expression
mbfmbfm	⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)

Proof of Theorem mbfmbfm

Step	Hyp	Ref	Expression
1		mbfmbfm.1	. 2 ⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))
2	1	isanmbfm 34440	1 ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)

Syntax hints:   → wi 4   ∈ wcel 2119  ∪ cuni 4838  dom cdm 5618  ran crn 5619  ‘cfv 6485  (class class class)co 7356  sigaGencsigagen 34322  MblFnMcmbfm 34433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-cnv 5626  df-dm 5628  df-rn 5629  df-iota 6441  df-fv 6493  df-ov 7359

This theorem is referenced by: (None)