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Theorem pimltmnf2 46653
Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
Hypotheses
Ref Expression
pimltmnf2.1 𝑥𝐹
pimltmnf2.2 (𝜑𝐹:𝐴⟶ℝ)
Assertion
Ref Expression
pimltmnf2 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < -∞} = ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem pimltmnf2
StepHypRef Expression
1 pimltmnf2.1 . 2 𝑥𝐹
2 nfcv 2902 . 2 𝑥𝐴
3 pimltmnf2.2 . 2 (𝜑𝐹:𝐴⟶ℝ)
41, 2, 3pimltmnf2f 46652 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < -∞} = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wnfc 2887  {crab 3432  c0 4338   class class class wbr 5147  wf 6558  cfv 6562  cr 11151  -∞cmnf 11290   < clt 11292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298
This theorem is referenced by: (None)
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