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Theorem pimgtmnf 46738
Description: Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 20-Dec-2024.)
Hypotheses
Ref Expression
pimgtmnf.1 𝑥𝜑
pimgtmnf.2 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
Assertion
Ref Expression
pimgtmnf (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem pimgtmnf
StepHypRef Expression
1 pimgtmnf.1 . 2 𝑥𝜑
2 nfcv 2905 . 2 𝑥𝐴
3 pimgtmnf.2 . 2 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
41, 2, 3pimgtmnff 46737 1 (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wnf 1783  wcel 2108  {crab 3436   class class class wbr 5143  cr 11154  -∞cmnf 11293   < clt 11295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300
This theorem is referenced by:  smfpimgtxr  46795
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