Theorem pimconstlt0 46083

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 7012	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
9	3, 8	breqtrrd 5170	. . . . 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 11288	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ*)
14	11, 13	xrlenltd 11304	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ≤ (𝐹‘𝑥) ↔ ¬ (𝐹‘𝑥) < 𝐶))
15	9, 14	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ (𝐹‘𝑥) < 𝐶)
16	15	ex 412	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ¬ (𝐹‘𝑥) < 𝐶))
17	1, 16	ralrimi 3250	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) < 𝐶)
18		rabeq0 4380	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) < 𝐶)
19	17, 18	sylibr 233	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝐶} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1534  Ⅎwnf 1778   ∈ wcel 2099  ∀wral 3057  {crab 3428  ∅c0 4318   class class class wbr 5142   ↦ cmpt 5225  ‘cfv 6542  ℝcr 11131  ℝ^*cxr 11271   < clt 11272   ≤ cle 11273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-xr 11276  df-le 11278

This theorem is referenced by:  smfconst  46131