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Theorem pimltpnff 45991
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 44381 . . 3 {𝑥𝐴𝐵 < +∞} ⊆ 𝐴
32a1i 11 . 2 (𝜑 → {𝑥𝐴𝐵 < +∞} ⊆ 𝐴)
4 pimltpnff.1 . . . 4 𝑥𝜑
5 simpr 484 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑥𝐴)
6 pimltpnff.3 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
7 ltpnf 13106 . . . . . . . 8 (𝐵 ∈ ℝ → 𝐵 < +∞)
86, 7syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 < +∞)
95, 8jca 511 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝐴𝐵 < +∞))
10 rabid 3446 . . . . . 6 (𝑥 ∈ {𝑥𝐴𝐵 < +∞} ↔ (𝑥𝐴𝐵 < +∞))
119, 10sylibr 233 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1211ex 412 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝐴𝐵 < +∞}))
134, 12ralrimi 3248 . . 3 (𝜑 → ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
14 nfrab1 3445 . . . 4 𝑥{𝑥𝐴𝐵 < +∞}
151, 14dfss3f 3968 . . 3 (𝐴 ⊆ {𝑥𝐴𝐵 < +∞} ↔ ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1613, 15sylibr 233 . 2 (𝜑𝐴 ⊆ {𝑥𝐴𝐵 < +∞})
173, 16eqssd 3994 1 (𝜑 → {𝑥𝐴𝐵 < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wnf 1777  wcel 2098  wnfc 2877  wral 3055  {crab 3426  wss 3943   class class class wbr 5141  cr 11111  +∞cpnf 11249   < clt 11252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-pnf 11254  df-xr 11256  df-ltxr 11257
This theorem is referenced by:  pimltpnf  45992
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