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Theorem mbfmcnvima 30293
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 30288 . . . 4 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
62, 5mpbid 222 . . 3 (𝜑 → (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
76simprd 479 . 2 (𝜑 → ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)
8 imaeq2 5450 . . . 4 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
98eleq1d 2684 . . 3 (𝑥 = 𝐴 → ((𝐹𝑥) ∈ 𝑆 ↔ (𝐹𝐴) ∈ 𝑆))
109rspcv 3300 . 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 384   = wceq 1481  wcel 1988  wral 2909   cuni 4427  ccnv 5103  ran crn 5105  cima 5107  (class class class)co 6635  𝑚 cmap 7842  sigAlgebracsiga 30144  MblFnMcmbfm 30286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-mbfm 30287
This theorem is referenced by:  imambfm  30298  mbfmco  30300  mbfmco2  30301  sxbrsiga  30326  sibfinima  30375  sibfof  30376  orvcoel  30497  orvccel  30498
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