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Theorem mbfmcnvima 34237
Description: The preimage by a measurable function is a measurable set. (Contributed by Thierry Arnoux, 23-Jan-2017.)
Hypotheses
Ref Expression
mbfmf.1 (𝜑𝑆 ran sigAlgebra)
mbfmf.2 (𝜑𝑇 ran sigAlgebra)
mbfmf.3 (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
mbfmcnvima.4 (𝜑𝐴𝑇)
Assertion
Ref Expression
mbfmcnvima (𝜑 → (𝐹𝐴) ∈ 𝑆)

Proof of Theorem mbfmcnvima
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 imaeq2 6076 . . 3 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
21eleq1d 2824 . 2 (𝑥 = 𝐴 → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹𝐴) ∈ 𝑆))
3 mbfmf.3 . . . 4 (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
4 mbfmf.1 . . . . 5 (𝜑𝑆 ran sigAlgebra)
5 mbfmf.2 . . . . 5 (𝜑𝑇 ran sigAlgebra)
64, 5ismbfm 34232 . . . 4 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
73, 6mpbid 232 . . 3 (𝜑 → (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
87simprd 495 . 2 (𝜑 → ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)
9 mbfmcnvima.4 . 2 (𝜑𝐴𝑇)
102, 8, 9rspcdva 3623 1 (𝜑 → (𝐹𝐴) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059   cuni 4912  ccnv 5688  ran crn 5690  cima 5692  (class class class)co 7431  m cmap 8865  sigAlgebracsiga 34089  MblFnMcmbfm 34230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-mbfm 34231
This theorem is referenced by:  imambfm  34244  mbfmco  34246  mbfmco2  34247  sxbrsiga  34272  sibfinima  34321  sibfof  34322  orvcoel  34443  orvccel  34444
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