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Theorem pimconstlt0 43181
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 484 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶𝐵)
4 pimconstlt0.f . . . . . . . 8 𝐹 = (𝑥𝐴𝐵)
54a1i 11 . . . . . . 7 (𝜑𝐹 = (𝑥𝐴𝐵))
6 pimconstlt0.b . . . . . . . 8 (𝜑𝐵 ∈ ℝ)
76adantr 484 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
85, 7fvmpt2d 6762 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
93, 8breqtrrd 5075 . . . . 5 ((𝜑𝑥𝐴) → 𝐶 ≤ (𝐹𝑥))
10 pimconstlt0.c . . . . . . 7 (𝜑𝐶 ∈ ℝ*)
1110adantr 484 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ*)
128, 7eqeltrd 2916 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ ℝ)
1312rexrd 10676 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ ℝ*)
1411, 13xrlenltd 10692 . . . . 5 ((𝜑𝑥𝐴) → (𝐶 ≤ (𝐹𝑥) ↔ ¬ (𝐹𝑥) < 𝐶))
159, 14mpbid 235 . . . 4 ((𝜑𝑥𝐴) → ¬ (𝐹𝑥) < 𝐶)
1615ex 416 . . 3 (𝜑 → (𝑥𝐴 → ¬ (𝐹𝑥) < 𝐶))
171, 16ralrimi 3210 . 2 (𝜑 → ∀𝑥𝐴 ¬ (𝐹𝑥) < 𝐶)
18 rabeq0 4319 . 2 ({𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = ∅ ↔ ∀𝑥𝐴 ¬ (𝐹𝑥) < 𝐶)
1917, 18sylibr 237 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wnf 1785  wcel 2115  wral 3132  {crab 3136  c0 4274   class class class wbr 5047  cmpt 5127  cfv 6336  cr 10521  *cxr 10659   < clt 10660  cle 10661
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fv 6344  df-xr 10664  df-le 10666
This theorem is referenced by:  smfconst  43225
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