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Theorem pimconstlt0 45407
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 481 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶𝐵)
4 pimconstlt0.f . . . . . . . 8 𝐹 = (𝑥𝐴𝐵)
54a1i 11 . . . . . . 7 (𝜑𝐹 = (𝑥𝐴𝐵))
6 pimconstlt0.b . . . . . . . 8 (𝜑𝐵 ∈ ℝ)
76adantr 481 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
85, 7fvmpt2d 7011 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
93, 8breqtrrd 5176 . . . . 5 ((𝜑𝑥𝐴) → 𝐶 ≤ (𝐹𝑥))
10 pimconstlt0.c . . . . . . 7 (𝜑𝐶 ∈ ℝ*)
1110adantr 481 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶 ∈ ℝ*)
128, 7eqeltrd 2833 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ ℝ)
1312rexrd 11263 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ ℝ*)
1411, 13xrlenltd 11279 . . . . 5 ((𝜑𝑥𝐴) → (𝐶 ≤ (𝐹𝑥) ↔ ¬ (𝐹𝑥) < 𝐶))
159, 14mpbid 231 . . . 4 ((𝜑𝑥𝐴) → ¬ (𝐹𝑥) < 𝐶)
1615ex 413 . . 3 (𝜑 → (𝑥𝐴 → ¬ (𝐹𝑥) < 𝐶))
171, 16ralrimi 3254 . 2 (𝜑 → ∀𝑥𝐴 ¬ (𝐹𝑥) < 𝐶)
18 rabeq0 4384 . 2 ({𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = ∅ ↔ ∀𝑥𝐴 ¬ (𝐹𝑥) < 𝐶)
1917, 18sylibr 233 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wnf 1785  wcel 2106  wral 3061  {crab 3432  c0 4322   class class class wbr 5148  cmpt 5231  cfv 6543  cr 11108  *cxr 11246   < clt 11247  cle 11248
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-xr 11251  df-le 11253
This theorem is referenced by:  smfconst  45455
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