Theorem pimltpnf 46709

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

Hypotheses

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

Assertion

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

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem pimltpnf

Step	Hyp	Ref	Expression
1		pimltpnf.1	. 2 ⊢ Ⅎ𝑥𝜑
2		nfcv 2892	. 2 ⊢ Ⅎ𝑥𝐴
3		pimltpnf.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	1, 2, 3	pimltpnff 46708	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  {crab 3408   class class class wbr 5110  ℝcr 11074  +∞cpnf 11212   < clt 11215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-xp 5647  df-pnf 11217  df-xr 11219  df-ltxr 11220

This theorem is referenced by:  pimltpnf2f  46717