Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pimconstlt0 Structured version   Visualization version   GIF version

Theorem pimconstlt0 41708
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 𝑥𝜑
pimconstlt0.b (𝜑𝐵 ∈ ℝ)
pimconstlt0.f 𝐹 = (𝑥𝐴𝐵)
pimconstlt0.c (𝜑𝐶 ∈ ℝ*)
pimconstlt0.l (𝜑𝐶𝐵)
Assertion
Ref Expression
pimconstlt0 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem pimconstlt0
StepHypRef Expression
1 pimconstlt0.x . . 3 𝑥𝜑
2 pimconstlt0.l . . . . . . 7 (𝜑𝐶𝐵)
32adantr 474 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶𝐵)
4 pimconstlt0.f . . . . . . . 8 𝐹 = (𝑥𝐴𝐵)
54a1i 11 . . . . . . 7 (𝜑𝐹 = (𝑥𝐴𝐵))
6 pimconstlt0.b . . . . . . . 8 (𝜑𝐵 ∈ ℝ)
76adantr 474 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
85, 7fvmpt2d 6540 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
93, 8breqtrrd 4901 . . . . 5 ((𝜑𝑥𝐴) → 𝐶 ≤ (𝐹𝑥))
10 pimconstlt0.c . . . . . . 7 (𝜑𝐶 ∈ ℝ*)
1110adantr 474 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ*)
128, 7eqeltrd 2906 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ ℝ)
1312rexrd 10406 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ ℝ*)
1411, 13xrlenltd 10423 . . . . 5 ((𝜑𝑥𝐴) → (𝐶 ≤ (𝐹𝑥) ↔ ¬ (𝐹𝑥) < 𝐶))
159, 14mpbid 224 . . . 4 ((𝜑𝑥𝐴) → ¬ (𝐹𝑥) < 𝐶)
1615ex 403 . . 3 (𝜑 → (𝑥𝐴 → ¬ (𝐹𝑥) < 𝐶))
171, 16ralrimi 3166 . 2 (𝜑 → ∀𝑥𝐴 ¬ (𝐹𝑥) < 𝐶)
18 rabeq0 4186 . 2 ({𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = ∅ ↔ ∀𝑥𝐴 ¬ (𝐹𝑥) < 𝐶)
1917, 18sylibr 226 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386   = wceq 1658  wnf 1884  wcel 2166  wral 3117  {crab 3121  c0 4144   class class class wbr 4873  cmpt 4952  cfv 6123  cr 10251  *cxr 10390   < clt 10391  cle 10392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fv 6131  df-xr 10395  df-le 10397
This theorem is referenced by:  smfconst  41752
  Copyright terms: Public domain W3C validator