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Theorem pimltpnff 46659
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, 20-Dec-2024.)
Hypotheses
Ref Expression
pimltpnff.1 𝑥𝜑
pimltpnff.2 𝑥𝐴
pimltpnff.3 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
Assertion
Ref Expression
pimltpnff (𝜑 → {𝑥𝐴𝐵 < +∞} = 𝐴)

Proof of Theorem pimltpnff
StepHypRef Expression
1 pimltpnff.2 . . . 4 𝑥𝐴
21ssrab2f 45057 . . 3 {𝑥𝐴𝐵 < +∞} ⊆ 𝐴
32a1i 11 . 2 (𝜑 → {𝑥𝐴𝐵 < +∞} ⊆ 𝐴)
4 pimltpnff.1 . . . 4 𝑥𝜑
5 simpr 484 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑥𝐴)
6 pimltpnff.3 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
7 ltpnf 13160 . . . . . . . 8 (𝐵 ∈ ℝ → 𝐵 < +∞)
86, 7syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 < +∞)
95, 8jca 511 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝐴𝐵 < +∞))
10 rabid 3455 . . . . . 6 (𝑥 ∈ {𝑥𝐴𝐵 < +∞} ↔ (𝑥𝐴𝐵 < +∞))
119, 10sylibr 234 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1211ex 412 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝐴𝐵 < +∞}))
134, 12ralrimi 3255 . . 3 (𝜑 → ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
14 nfrab1 3454 . . . 4 𝑥{𝑥𝐴𝐵 < +∞}
151, 14dfss3f 3987 . . 3 (𝐴 ⊆ {𝑥𝐴𝐵 < +∞} ↔ ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1613, 15sylibr 234 . 2 (𝜑𝐴 ⊆ {𝑥𝐴𝐵 < +∞})
173, 16eqssd 4013 1 (𝜑 → {𝑥𝐴𝐵 < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wnf 1780  wcel 2106  wnfc 2888  wral 3059  {crab 3433  wss 3963   class class class wbr 5148  cr 11152  +∞cpnf 11290   < clt 11293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-pnf 11295  df-xr 11297  df-ltxr 11298
This theorem is referenced by:  pimltpnf  46660
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