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Theorem smfpreimaltf 46183
Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
smfpreimaltf.x β„²π‘₯𝐹
smfpreimaltf.s (πœ‘ β†’ 𝑆 ∈ SAlg)
smfpreimaltf.f (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))
smfpreimaltf.d 𝐷 = dom 𝐹
smfpreimaltf.a (πœ‘ β†’ 𝐴 ∈ ℝ)
Assertion
Ref Expression
smfpreimaltf (πœ‘ β†’ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < 𝐴} ∈ (𝑆 β†Ύt 𝐷))
Distinct variable group:   π‘₯,𝐴
Allowed substitution hints:   πœ‘(π‘₯)   𝐷(π‘₯)   𝑆(π‘₯)   𝐹(π‘₯)

Proof of Theorem smfpreimaltf
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 smfpreimaltf.a . 2 (πœ‘ β†’ 𝐴 ∈ ℝ)
2 smfpreimaltf.f . . . 4 (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))
3 smfpreimaltf.x . . . . 5 β„²π‘₯𝐹
4 smfpreimaltf.s . . . . 5 (πœ‘ β†’ 𝑆 ∈ SAlg)
5 smfpreimaltf.d . . . . 5 𝐷 = dom 𝐹
63, 4, 5issmff 46181 . . . 4 (πœ‘ β†’ (𝐹 ∈ (SMblFnβ€˜π‘†) ↔ (𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘Ž ∈ ℝ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷))))
72, 6mpbid 231 . . 3 (πœ‘ β†’ (𝐷 βŠ† βˆͺ 𝑆 ∧ 𝐹:π·βŸΆβ„ ∧ βˆ€π‘Ž ∈ ℝ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷)))
87simp3d 1141 . 2 (πœ‘ β†’ βˆ€π‘Ž ∈ ℝ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷))
9 breq2 5148 . . . . 5 (π‘Ž = 𝐴 β†’ ((πΉβ€˜π‘₯) < π‘Ž ↔ (πΉβ€˜π‘₯) < 𝐴))
109rabbidv 3427 . . . 4 (π‘Ž = 𝐴 β†’ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} = {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < 𝐴})
1110eleq1d 2810 . . 3 (π‘Ž = 𝐴 β†’ ({π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷) ↔ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < 𝐴} ∈ (𝑆 β†Ύt 𝐷)))
1211rspcva 3601 . 2 ((𝐴 ∈ ℝ ∧ βˆ€π‘Ž ∈ ℝ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt 𝐷)) β†’ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < 𝐴} ∈ (𝑆 β†Ύt 𝐷))
131, 8, 12syl2anc 582 1 (πœ‘ β†’ {π‘₯ ∈ 𝐷 ∣ (πΉβ€˜π‘₯) < 𝐴} ∈ (𝑆 β†Ύt 𝐷))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  β„²wnfc 2875  βˆ€wral 3051  {crab 3419   βŠ† wss 3941  βˆͺ cuni 4904   class class class wbr 5144  dom cdm 5673  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7413  β„cr 11132   < clt 11273   β†Ύt crest 17396  SAlgcsalg 45755  SMblFncsmblfn 46142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-pre-lttri 11207  ax-pre-lttrn 11208
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-po 5585  df-so 5586  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7987  df-2nd 7988  df-er 8718  df-pm 8841  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-ioo 13355  df-ico 13357  df-smblfn 46143
This theorem is referenced by:  smfpimltmpt  46193  smfpimltxr  46194
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