Theorem pimgtmnf 42994

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.)

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	. . 3 ⊢ Ⅎ𝑥𝜑
2		eqidd 2822	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
3		pimgtmnf.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	2, 3	fvmpt2d 6775	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
5	4	eqcomd 2827	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
6	5	breq2d 5070	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-∞ < 𝐵 ↔ -∞ < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
7	1, 6	rabbida 3474	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = {𝑥 ∈ 𝐴 ∣ -∞ < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)})
8		nfmpt1 5156	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9		eqid 2821	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
10	1, 3, 9	fmptdf 6875	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
11	8, 10	pimgtmnf2 42986	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)} = 𝐴)
12	7, 11	eqtrd 2856	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533  Ⅎwnf 1780   ∈ wcel 2110  {crab 3142   class class class wbr 5058   ↦ cmpt 5138  ‘cfv 6349  ℝcr 10530  -∞cmnf 10667   < clt 10669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674

This theorem is referenced by:  smfpimgtxr  43050