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Theorem pimltpnf 41422
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 3828 . . 3 {𝑥𝐴𝐵 < +∞} ⊆ 𝐴
21a1i 11 . 2 (𝜑 → {𝑥𝐴𝐵 < +∞} ⊆ 𝐴)
3 pimltpnf.1 . . . 4 𝑥𝜑
4 simpr 479 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑥𝐴)
5 pimltpnf.2 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
6 ltpnf 12147 . . . . . . . 8 (𝐵 ∈ ℝ → 𝐵 < +∞)
75, 6syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 < +∞)
84, 7jca 555 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝐴𝐵 < +∞))
9 rabid 3254 . . . . . 6 (𝑥 ∈ {𝑥𝐴𝐵 < +∞} ↔ (𝑥𝐴𝐵 < +∞))
108, 9sylibr 224 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1110ex 449 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝐴𝐵 < +∞}))
123, 11ralrimi 3095 . . 3 (𝜑 → ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
13 nfcv 2902 . . . 4 𝑥𝐴
14 nfrab1 3261 . . . 4 𝑥{𝑥𝐴𝐵 < +∞}
1513, 14dfss3f 3736 . . 3 (𝐴 ⊆ {𝑥𝐴𝐵 < +∞} ↔ ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1612, 15sylibr 224 . 2 (𝜑𝐴 ⊆ {𝑥𝐴𝐵 < +∞})
172, 16eqssd 3761 1 (𝜑 → {𝑥𝐴𝐵 < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wnf 1857  wcel 2139  wral 3050  {crab 3054  wss 3715   class class class wbr 4804  cr 10127  +∞cpnf 10263   < clt 10266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-pnf 10268  df-xr 10270  df-ltxr 10271
This theorem is referenced by:  pimltpnf2  41429
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