Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pimltpnff Structured version   Visualization version   GIF version

Theorem pimltpnff 45405
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 43791 . . 3 {𝑥𝐴𝐵 < +∞} ⊆ 𝐴
32a1i 11 . 2 (𝜑 → {𝑥𝐴𝐵 < +∞} ⊆ 𝐴)
4 pimltpnff.1 . . . 4 𝑥𝜑
5 simpr 485 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑥𝐴)
6 pimltpnff.3 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
7 ltpnf 13096 . . . . . . . 8 (𝐵 ∈ ℝ → 𝐵 < +∞)
86, 7syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 < +∞)
95, 8jca 512 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝐴𝐵 < +∞))
10 rabid 3452 . . . . . 6 (𝑥 ∈ {𝑥𝐴𝐵 < +∞} ↔ (𝑥𝐴𝐵 < +∞))
119, 10sylibr 233 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1211ex 413 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝐴𝐵 < +∞}))
134, 12ralrimi 3254 . . 3 (𝜑 → ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
14 nfrab1 3451 . . . 4 𝑥{𝑥𝐴𝐵 < +∞}
151, 14dfss3f 3972 . . 3 (𝐴 ⊆ {𝑥𝐴𝐵 < +∞} ↔ ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴𝐵 < +∞})
1613, 15sylibr 233 . 2 (𝜑𝐴 ⊆ {𝑥𝐴𝐵 < +∞})
173, 16eqssd 3998 1 (𝜑 → {𝑥𝐴𝐵 < +∞} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wnf 1785  wcel 2106  wnfc 2883  wral 3061  {crab 3432  wss 3947   class class class wbr 5147  cr 11105  +∞cpnf 11241   < clt 11244
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-pnf 11246  df-xr 11248  df-ltxr 11249
This theorem is referenced by:  pimltpnf  45406
  Copyright terms: Public domain W3C validator