Theorem pimgtmnf 47003

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, 26-Jun-2021.) (Revised by Glauco Siliprandi, 20-Dec-2024.)

Hypotheses

Ref	Expression
pimgtmnf.1	⊢ Ⅎ𝑥𝜑
pimgtmnf.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)

Assertion

Ref	Expression
pimgtmnf	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem pimgtmnf

Step	Hyp	Ref	Expression
1		pimgtmnf.1	. 2 ⊢ Ⅎ𝑥𝜑
2		nfcv 2899	. 2 ⊢ Ⅎ𝑥𝐴
3		pimgtmnf.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	1, 2, 3	pimgtmnff 47002	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  {crab 3400   class class class wbr 5099  ℝcr 11029  -∞cmnf 11168   < clt 11170

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-xp 5631  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175

This theorem is referenced by:  smfpimgtxr  47060