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Theorem mbfmcnvima 29646
 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 mbfmcnvima.4 . 2 (𝜑𝐴𝑇)
2 mbfmf.3 . . . 4 (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
3 mbfmf.1 . . . . 5 (𝜑𝑆 ran sigAlgebra)
4 mbfmf.2 . . . . 5 (𝜑𝑇 ran sigAlgebra)
53, 4ismbfm 29641 . . . 4 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
62, 5mpbid 221 . . 3 (𝜑 → (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
76simprd 478 . 2 (𝜑 → ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)
8 imaeq2 5381 . . . 4 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
98eleq1d 2672 . . 3 (𝑥 = 𝐴 → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹𝐴) ∈ 𝑆))
109rspcv 3278 . 2 (𝐴𝑇 → (∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆 → (𝐹𝐴) ∈ 𝑆))
111, 7, 10sylc 63 1 (𝜑 → (𝐹𝐴) ∈ 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∪ cuni 4372  ◡ccnv 5037  ran crn 5039   “ cima 5041  (class class class)co 6549   ↑𝑚 cmap 7744  sigAlgebracsiga 29497  MblFnMcmbfm 29639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-mbfm 29640 This theorem is referenced by:  imambfm  29651  mbfmco  29653  mbfmco2  29654  sxbrsiga  29679  sibfinima  29728  sibfof  29729  orvcoel  29850  orvccel  29851
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