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Theorem pimgtmnff 46645
Description: Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 20-Dec-2024.)
Hypotheses
Ref Expression
pimgtmnff.1 𝑥𝜑
pimgtmnff.2 𝑥𝐴
pimgtmnff.3 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
Assertion
Ref Expression
pimgtmnff (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} = 𝐴)

Proof of Theorem pimgtmnff
StepHypRef Expression
1 pimgtmnff.2 . . . 4 𝑥𝐴
21ssrab2f 45021 . . 3 {𝑥𝐴 ∣ -∞ < 𝐵} ⊆ 𝐴
32a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} ⊆ 𝐴)
4 pimgtmnff.1 . . . 4 𝑥𝜑
5 simpr 484 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑥𝐴)
6 pimgtmnff.3 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
7 mnflt 13188 . . . . . . . 8 (𝐵 ∈ ℝ → -∞ < 𝐵)
86, 7syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → -∞ < 𝐵)
95, 8jca 511 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝐴 ∧ -∞ < 𝐵))
10 rabid 3465 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ -∞ < 𝐵} ↔ (𝑥𝐴 ∧ -∞ < 𝐵))
119, 10sylibr 234 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ {𝑥𝐴 ∣ -∞ < 𝐵})
1211ex 412 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝐴 ∣ -∞ < 𝐵}))
134, 12ralrimi 3263 . . 3 (𝜑 → ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴 ∣ -∞ < 𝐵})
14 nfrab1 3464 . . . 4 𝑥{𝑥𝐴 ∣ -∞ < 𝐵}
151, 14dfss3f 4000 . . 3 (𝐴 ⊆ {𝑥𝐴 ∣ -∞ < 𝐵} ↔ ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴 ∣ -∞ < 𝐵})
1613, 15sylibr 234 . 2 (𝜑𝐴 ⊆ {𝑥𝐴 ∣ -∞ < 𝐵})
173, 16eqssd 4026 1 (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wnf 1781  wcel 2108  wnfc 2893  wral 3067  {crab 3443  wss 3976   class class class wbr 5166  cr 11185  -∞cmnf 11324   < clt 11326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772  ax-cnex 11242
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-pnf 11328  df-mnf 11329  df-xr 11330  df-ltxr 11331
This theorem is referenced by:  pimgtmnf  46646
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