Theorem pimconstlt0 43339

Description: Given a constant function, its preimage with respect to an unbounded below, open interval, with upper bound less than or equal to the constant, is the empty set. Second part of Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
pimconstlt0.x	⊢ Ⅎ𝑥𝜑
pimconstlt0.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
pimconstlt0.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
pimconstlt0.c	⊢ (𝜑 → 𝐶 ∈ ℝ*)
pimconstlt0.l	⊢ (𝜑 → 𝐶 ≤ 𝐵)

Assertion

Ref	Expression
pimconstlt0	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅)

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem pimconstlt0

Step	Hyp	Ref	Expression
1		pimconstlt0.x	. . 3 ⊢ Ⅎ𝑥𝜑
2		pimconstlt0.l	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≤ 𝐵)
3	2	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ 𝐵)
4		pimconstlt0.f	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
6		pimconstlt0.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
7	6	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
8	5, 7	fvmpt2d 6758	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
9	3, 8	breqtrrd 5058	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ (𝐹‘𝑥))
10		pimconstlt0.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
11	10	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ*)
12	8, 7	eqeltrd 2890	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ)
13	12	rexrd 10680	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ*)
14	11, 13	xrlenltd 10696	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ≤ (𝐹‘𝑥) ↔ ¬ (𝐹‘𝑥) < 𝐶))
15	9, 14	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ (𝐹‘𝑥) < 𝐶)
16	15	ex 416	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ¬ (𝐹‘𝑥) < 𝐶))
17	1, 16	ralrimi 3180	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) < 𝐶)
18		rabeq0 4292	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) < 𝐶)
19	17, 18	sylibr 237	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  ∀wral 3106  {crab 3110  ∅c0 4243   class class class wbr 5030   ↦ cmpt 5110  ‘cfv 6324  ℝcr 10525  ℝ^*cxr 10663   < clt 10664   ≤ cle 10665

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

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-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332  df-xr 10668  df-le 10670

This theorem is referenced by:  smfconst  43383