Theorem pimltpnf 47401

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 2925	. 2 ⊢ Ⅎ𝑥𝐴
3		pimltpnf.2	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	1, 2, 3	pimltpnff 47400	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  Ⅎwnf 1813   ∈ wcel 2143  {crab 3416   class class class wbr 5110  ℝcr 11100  +∞cpnf 11241   < clt 11244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5669  df-pnf 11246  df-xr 11248  df-ltxr 11249

This theorem is referenced by:  pimltpnf2f  47409