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Theorem pimconstlt0 41428
Description: Given a constant function, its preimage with respect to an unbounded below, open interval, with upper bound smaller 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 466 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶𝐵)
4 pimconstlt0.f . . . . . . . 8 𝐹 = (𝑥𝐴𝐵)
54a1i 11 . . . . . . 7 (𝜑𝐹 = (𝑥𝐴𝐵))
6 pimconstlt0.b . . . . . . . 8 (𝜑𝐵 ∈ ℝ)
76adantr 466 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
85, 7fvmpt2d 6435 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
93, 8breqtrrd 4812 . . . . 5 ((𝜑𝑥𝐴) → 𝐶 ≤ (𝐹𝑥))
10 pimconstlt0.c . . . . . . 7 (𝜑𝐶 ∈ ℝ*)
1110adantr 466 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ*)
128, 7eqeltrd 2849 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ ℝ)
1312rexrd 10290 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ ℝ*)
1411, 13xrlenltd 10305 . . . . 5 ((𝜑𝑥𝐴) → (𝐶 ≤ (𝐹𝑥) ↔ ¬ (𝐹𝑥) < 𝐶))
159, 14mpbid 222 . . . 4 ((𝜑𝑥𝐴) → ¬ (𝐹𝑥) < 𝐶)
1615ex 397 . . 3 (𝜑 → (𝑥𝐴 → ¬ (𝐹𝑥) < 𝐶))
171, 16ralrimi 3105 . 2 (𝜑 → ∀𝑥𝐴 ¬ (𝐹𝑥) < 𝐶)
18 rabeq0 4101 . 2 ({𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = ∅ ↔ ∀𝑥𝐴 ¬ (𝐹𝑥) < 𝐶)
1917, 18sylibr 224 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1630  wnf 1855  wcel 2144  wral 3060  {crab 3064  c0 4061   class class class wbr 4784  cmpt 4861  cfv 6031  cr 10136  *cxr 10274   < clt 10275  cle 10276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-xr 10279  df-le 10281
This theorem is referenced by:  smfconst  41472
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