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Theorem pimgtmnf 46344
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 2892 . 2 𝑥𝐴
3 pimgtmnf.2 . 2 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
41, 2, 3pimgtmnff 46343 1 (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wnf 1778  wcel 2099  {crab 3419   class class class wbr 5153  cr 11157  -∞cmnf 11296   < clt 11298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-xp 5688  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303
This theorem is referenced by:  smfpimgtxr  46401
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