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Theorem pimltpnf 43280
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 4031 . . 3 {𝑥𝐴𝐵 < +∞} ⊆ 𝐴
21a1i 11 . 2 (𝜑 → {𝑥𝐴𝐵 < +∞} ⊆ 𝐴)
3 pimltpnf.1 . . . 4 𝑥𝜑
4 simpr 488 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑥𝐴)
5 pimltpnf.2 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
6 ltpnf 12503 . . . . . . . 8 (𝐵 ∈ ℝ → 𝐵 < +∞)
75, 6syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 < +∞)
84, 7jca 515 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝐴𝐵 < +∞))
9 rabid 3359 . . . . . 6 (𝑥 ∈ {𝑥𝐴𝐵 < +∞} ↔ (𝑥𝐴𝐵 < +∞))
108, 9sylibr 237 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1110ex 416 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝐴𝐵 < +∞}))
123, 11ralrimi 3205 . . 3 (𝜑 → ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
13 nfcv 2979 . . . 4 𝑥𝐴
14 nfrab1 3365 . . . 4 𝑥{𝑥𝐴𝐵 < +∞}
1513, 14dfss3f 3933 . . 3 (𝐴 ⊆ {𝑥𝐴𝐵 < +∞} ↔ ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1612, 15sylibr 237 . 2 (𝜑𝐴 ⊆ {𝑥𝐴𝐵 < +∞})
172, 16eqssd 3959 1 (𝜑 → {𝑥𝐴𝐵 < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wnf 1785  wcel 2114  wral 3130  {crab 3134  wss 3908   class class class wbr 5042  cr 10525  +∞cpnf 10661   < clt 10664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-xp 5538  df-pnf 10666  df-xr 10668  df-ltxr 10669
This theorem is referenced by:  pimltpnf2  43287
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