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Theorem mbfmcnvima 31414
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 5918 . . 3 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
21eleq1d 2894 . 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 31409 . . . 4 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
73, 6mpbid 233 . . 3 (𝜑 → (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
87simprd 496 . 2 (𝜑 → ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)
9 mbfmcnvima.4 . 2 (𝜑𝐴𝑇)
102, 8, 9rspcdva 3622 1 (𝜑 → (𝐹𝐴) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135   cuni 4830  ccnv 5547  ran crn 5549  cima 5551  (class class class)co 7145  m cmap 8395  sigAlgebracsiga 31266  MblFnMcmbfm 31407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-mbfm 31408
This theorem is referenced by:  imambfm  31419  mbfmco  31421  mbfmco2  31422  sxbrsiga  31447  sibfinima  31496  sibfof  31497  orvcoel  31618  orvccel  31619
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