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Theorem pimltpnf2 45202
Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
Hypotheses
Ref Expression
pimltpnf2.1 𝑥𝐹
pimltpnf2.2 (𝜑𝐹:𝐴⟶ℝ)
Assertion
Ref Expression
pimltpnf2 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem pimltpnf2
StepHypRef Expression
1 pimltpnf2.1 . 2 𝑥𝐹
2 nfcv 2902 . 2 𝑥𝐴
3 pimltpnf2.2 . 2 (𝜑𝐹:𝐴⟶ℝ)
41, 2, 3pimltpnf2f 45201 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wnfc 2882  {crab 3431   class class class wbr 5141  wf 6528  cfv 6532  cr 11091  +∞cpnf 11227   < clt 11230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-pnf 11232  df-xr 11234  df-ltxr 11235
This theorem is referenced by: (None)
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