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Theorem mbfmcnvima 32895
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 6014 . . 3 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
21eleq1d 2823 . 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 32890 . . . 4 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
73, 6mpbid 231 . . 3 (𝜑 → (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
87simprd 497 . 2 (𝜑 → ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)
9 mbfmcnvima.4 . 2 (𝜑𝐴𝑇)
102, 8, 9rspcdva 3585 1 (𝜑 → (𝐹𝐴) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3065   cuni 4870  ccnv 5637  ran crn 5639  cima 5641  (class class class)co 7362  m cmap 8772  sigAlgebracsiga 32747  MblFnMcmbfm 32888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-mbfm 32889
This theorem is referenced by:  imambfm  32902  mbfmco  32904  mbfmco2  32905  sxbrsiga  32930  sibfinima  32979  sibfof  32980  orvcoel  33101  orvccel  33102
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