Theorem pimgtmnf 39609

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 2611	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
3		pimgtmnf.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4	2, 3	fvmpt2d 6202	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
5	4	eqcomd 2616	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
6	5	breq2d 4595	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (-∞ < 𝐵 ↔ -∞ < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))
7	1, 6	rabbida 38302	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = {𝑥 ∈ 𝐴 ∣ -∞ < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)})
8		nfmpt1 4675	. . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
9		eqid 2610	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
10	1, 3, 9	fmptdf 6294	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
11	8, 10	pimgtmnf2 39601	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)} = 𝐴)
12	7, 11	eqtrd 2644	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  {crab 2900   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  ℝcr 9814  -∞cmnf 9951   < clt 9953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958

This theorem is referenced by:  smfpimgtxr  39666