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Theorem pimltpnf 39387
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.)
Hypotheses
Ref Expression
pimltpnf.1 𝑥𝜑
pimltpnf.2 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
Assertion
Ref Expression
pimltpnf (𝜑 → {𝑥𝐴𝐵 < +∞} = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem pimltpnf
StepHypRef Expression
1 ssrab2 3650 . . 3 {𝑥𝐴𝐵 < +∞} ⊆ 𝐴
21a1i 11 . 2 (𝜑 → {𝑥𝐴𝐵 < +∞} ⊆ 𝐴)
3 pimltpnf.1 . . . 4 𝑥𝜑
4 simpr 476 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑥𝐴)
5 pimltpnf.2 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
6 ltpnf 11794 . . . . . . . 8 (𝐵 ∈ ℝ → 𝐵 < +∞)
75, 6syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 < +∞)
84, 7jca 553 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝐴𝐵 < +∞))
9 rabid 3095 . . . . . 6 (𝑥 ∈ {𝑥𝐴𝐵 < +∞} ↔ (𝑥𝐴𝐵 < +∞))
108, 9sylibr 223 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1110ex 449 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝐴𝐵 < +∞}))
123, 11ralrimi 2940 . . 3 (𝜑 → ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
13 nfcv 2751 . . . 4 𝑥𝐴
14 nfrab1 3099 . . . 4 𝑥{𝑥𝐴𝐵 < +∞}
1513, 14dfss3f 3560 . . 3 (𝐴 ⊆ {𝑥𝐴𝐵 < +∞} ↔ ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1612, 15sylibr 223 . 2 (𝜑𝐴 ⊆ {𝑥𝐴𝐵 < +∞})
172, 16eqssd 3585 1 (𝜑 → {𝑥𝐴𝐵 < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wnf 1699  wcel 1977  wral 2896  {crab 2900  wss 3540   class class class wbr 4578  cr 9792  +∞cpnf 9928   < clt 9931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-xp 5034  df-pnf 9933  df-xr 9935  df-ltxr 9936
This theorem is referenced by:  pimltpnf2  39394
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