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Theorem pimgtmnf 46678
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 2902 . 2 𝑥𝐴
3 pimgtmnf.2 . 2 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
41, 2, 3pimgtmnff 46677 1 (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wnf 1779  wcel 2105  {crab 3432   class class class wbr 5147  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
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-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  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-xp 5694  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297
This theorem is referenced by:  smfpimgtxr  46735
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