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Theorem pimltpnff 45030
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 43415 . . 3 {𝑥𝐴𝐵 < +∞} ⊆ 𝐴
32a1i 11 . 2 (𝜑 → {𝑥𝐴𝐵 < +∞} ⊆ 𝐴)
4 pimltpnff.1 . . . 4 𝑥𝜑
5 simpr 486 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑥𝐴)
6 pimltpnff.3 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
7 ltpnf 13046 . . . . . . . 8 (𝐵 ∈ ℝ → 𝐵 < +∞)
86, 7syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 < +∞)
95, 8jca 513 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝐴𝐵 < +∞))
10 rabid 3426 . . . . . 6 (𝑥 ∈ {𝑥𝐴𝐵 < +∞} ↔ (𝑥𝐴𝐵 < +∞))
119, 10sylibr 233 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1211ex 414 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝐴𝐵 < +∞}))
134, 12ralrimi 3239 . . 3 (𝜑 → ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
14 nfrab1 3425 . . . 4 𝑥{𝑥𝐴𝐵 < +∞}
151, 14dfss3f 3936 . . 3 (𝐴 ⊆ {𝑥𝐴𝐵 < +∞} ↔ ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1613, 15sylibr 233 . 2 (𝜑𝐴 ⊆ {𝑥𝐴𝐵 < +∞})
173, 16eqssd 3962 1 (𝜑 → {𝑥𝐴𝐵 < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wnf 1786  wcel 2107  wnfc 2884  wral 3061  {crab 3406  wss 3911   class class class wbr 5106  cr 11055  +∞cpnf 11191   < clt 11194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-pnf 11196  df-xr 11198  df-ltxr 11199
This theorem is referenced by:  pimltpnf  45031
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