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Theorem mbfmcnvima 30835
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 30830 . . . 4 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
62, 5mpbid 224 . . 3 (𝜑 → (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
76simprd 490 . 2 (𝜑 → ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)
8 imaeq2 5679 . . . 4 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
98eleq1d 2863 . . 3 (𝑥 = 𝐴 → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹𝐴) ∈ 𝑆))
109rspcv 3493 . 2 (𝐴𝑇 → (∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆 → (𝐹𝐴) ∈ 𝑆))
111, 7, 10sylc 65 1 (𝜑 → (𝐹𝐴) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089   cuni 4628  ccnv 5311  ran crn 5313  cima 5315  (class class class)co 6878  𝑚 cmap 8095  sigAlgebracsiga 30686  MblFnMcmbfm 30828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-mbfm 30829
This theorem is referenced by:  imambfm  30840  mbfmco  30842  mbfmco2  30843  sxbrsiga  30868  sibfinima  30917  sibfof  30918  orvcoel  31040  orvccel  31041
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