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Theorem pimconstlt1 46658
Description: Given a constant function, its preimage with respect to an unbounded below, open interval, with upper bound larger than the constant, is the whole domain. First part of Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
pimconstlt1.1 𝑥𝜑
pimconstlt1.2 (𝜑𝐵 ∈ ℝ)
pimconstlt1.3 𝐹 = (𝑥𝐴𝐵)
pimconstlt1.4 (𝜑𝐵 < 𝐶)
Assertion
Ref Expression
pimconstlt1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem pimconstlt1
StepHypRef Expression
1 ssrab2 4090 . . 3 {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ⊆ 𝐴
21a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ⊆ 𝐴)
3 pimconstlt1.1 . . . 4 𝑥𝜑
4 simpr 484 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑥𝐴)
5 pimconstlt1.3 . . . . . . . . . 10 𝐹 = (𝑥𝐴𝐵)
65a1i 11 . . . . . . . . 9 (𝜑𝐹 = (𝑥𝐴𝐵))
7 pimconstlt1.2 . . . . . . . . . 10 (𝜑𝐵 ∈ ℝ)
87adantr 480 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
96, 8fvmpt2d 7029 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
10 pimconstlt1.4 . . . . . . . . 9 (𝜑𝐵 < 𝐶)
1110adantr 480 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 < 𝐶)
129, 11eqbrtrd 5170 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) < 𝐶)
134, 12jca 511 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝐴 ∧ (𝐹𝑥) < 𝐶))
14 rabid 3455 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ↔ (𝑥𝐴 ∧ (𝐹𝑥) < 𝐶))
1513, 14sylibr 234 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
1615ex 412 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶}))
173, 16ralrimi 3255 . . 3 (𝜑 → ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
18 nfcv 2903 . . . 4 𝑥𝐴
19 nfrab1 3454 . . . 4 𝑥{𝑥𝐴 ∣ (𝐹𝑥) < 𝐶}
2018, 19dfss3f 3987 . . 3 (𝐴 ⊆ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ↔ ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
2117, 20sylibr 234 . 2 (𝜑𝐴 ⊆ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
222, 21eqssd 4013 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wnf 1780  wcel 2106  wral 3059  {crab 3433  wss 3963   class class class wbr 5148  cmpt 5231  cfv 6563  cr 11152   < clt 11293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by:  smfconst  46705
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