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Theorem pimltpnf 42445
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 3941 . . 3 {𝑥𝐴𝐵 < +∞} ⊆ 𝐴
21a1i 11 . 2 (𝜑 → {𝑥𝐴𝐵 < +∞} ⊆ 𝐴)
3 pimltpnf.1 . . . 4 𝑥𝜑
4 simpr 477 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑥𝐴)
5 pimltpnf.2 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
6 ltpnf 12331 . . . . . . . 8 (𝐵 ∈ ℝ → 𝐵 < +∞)
75, 6syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 < +∞)
84, 7jca 504 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝐴𝐵 < +∞))
9 rabid 3312 . . . . . 6 (𝑥 ∈ {𝑥𝐴𝐵 < +∞} ↔ (𝑥𝐴𝐵 < +∞))
108, 9sylibr 226 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1110ex 405 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝐴𝐵 < +∞}))
123, 11ralrimi 3161 . . 3 (𝜑 → ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
13 nfcv 2927 . . . 4 𝑥𝐴
14 nfrab1 3319 . . . 4 𝑥{𝑥𝐴𝐵 < +∞}
1513, 14dfss3f 3845 . . 3 (𝐴 ⊆ {𝑥𝐴𝐵 < +∞} ↔ ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1612, 15sylibr 226 . 2 (𝜑𝐴 ⊆ {𝑥𝐴𝐵 < +∞})
172, 16eqssd 3870 1 (𝜑 → {𝑥𝐴𝐵 < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1508  wnf 1747  wcel 2051  wral 3083  {crab 3087  wss 3824   class class class wbr 4926  cr 10333  +∞cpnf 10470   < clt 10473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-cnex 10390
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-xp 5410  df-pnf 10475  df-xr 10477  df-ltxr 10478
This theorem is referenced by:  pimltpnf2  42452
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