Theorem pimltpnf 43341

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.)

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		ssrab2 4007	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} ⊆ 𝐴
2	1	a1i 11	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} ⊆ 𝐴)
3		pimltpnf.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
5		pimltpnf.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
6		ltpnf 12503	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
7	5, 6	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 < +∞)
8	4, 7	jca 515	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ 𝐵 < +∞))
9		rabid 3331	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 < +∞))
10	8, 9	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞})
11	10	ex 416	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞}))
12	3, 11	ralrimi 3180	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞})
13		nfcv 2955	. . . 4 ⊢ Ⅎ𝑥𝐴
14		nfrab1 3337	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐵 < +∞}
15	13, 14	dfss3f 3906	. . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞})
16	12, 15	sylibr 237	. 2 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞})
17	2, 16	eqssd 3932	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  ∀wral 3106  {crab 3110   ⊆ wss 3881   class class class wbr 5030  ℝcr 10525  +∞cpnf 10661   < clt 10664

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-pnf 10666  df-xr 10668  df-ltxr 10669

This theorem is referenced by:  pimltpnf2  43348