Theorem pimconstlt0 46706

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 480	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
8	5, 7	fvmpt2d 6984	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
9	3, 8	breqtrrd 5138	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ (𝐹‘𝑥))
10		pimconstlt0.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ*)
12	8, 7	eqeltrd 2829	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ)
13	12	rexrd 11231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ*)
14	11, 13	xrlenltd 11247	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ≤ (𝐹‘𝑥) ↔ ¬ (𝐹‘𝑥) < 𝐶))
15	9, 14	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ (𝐹‘𝑥) < 𝐶)
16	15	ex 412	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ¬ (𝐹‘𝑥) < 𝐶))
17	1, 16	ralrimi 3236	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) < 𝐶)
18		rabeq0 4354	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) < 𝐶)
19	17, 18	sylibr 234	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  ∀wral 3045  {crab 3408  ∅c0 4299   class class class wbr 5110   ↦ cmpt 5191  ‘cfv 6514  ℝcr 11074  ℝ^*cxr 11214   < clt 11215   ≤ cle 11216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fv 6522  df-xr 11219  df-le 11221

This theorem is referenced by:  smfconst  46754