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Theorem pimgtmnff 46173
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 44548 . . 3 {𝑥𝐴 ∣ -∞ < 𝐵} ⊆ 𝐴
32a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} ⊆ 𝐴)
4 pimgtmnff.1 . . . 4 𝑥𝜑
5 simpr 483 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑥𝐴)
6 pimgtmnff.3 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
7 mnflt 13135 . . . . . . . 8 (𝐵 ∈ ℝ → -∞ < 𝐵)
86, 7syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → -∞ < 𝐵)
95, 8jca 510 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝐴 ∧ -∞ < 𝐵))
10 rabid 3440 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ -∞ < 𝐵} ↔ (𝑥𝐴 ∧ -∞ < 𝐵))
119, 10sylibr 233 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ {𝑥𝐴 ∣ -∞ < 𝐵})
1211ex 411 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝐴 ∣ -∞ < 𝐵}))
134, 12ralrimi 3245 . . 3 (𝜑 → ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴 ∣ -∞ < 𝐵})
14 nfrab1 3439 . . . 4 𝑥{𝑥𝐴 ∣ -∞ < 𝐵}
151, 14dfss3f 3963 . . 3 (𝐴 ⊆ {𝑥𝐴 ∣ -∞ < 𝐵} ↔ ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴 ∣ -∞ < 𝐵})
1613, 15sylibr 233 . 2 (𝜑𝐴 ⊆ {𝑥𝐴 ∣ -∞ < 𝐵})
173, 16eqssd 3990 1 (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wnf 1777  wcel 2098  wnfc 2875  wral 3051  {crab 3419  wss 3939   class class class wbr 5143  cr 11137  -∞cmnf 11276   < clt 11278
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-xp 5678  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283
This theorem is referenced by:  pimgtmnf  46174
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