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Theorem pimconstlt1 41661
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 3883 . . 3 {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ⊆ 𝐴
21a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ⊆ 𝐴)
3 pimconstlt1.1 . . . 4 𝑥𝜑
4 simpr 478 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑥𝐴)
5 pimconstlt1.3 . . . . . . . . . 10 𝐹 = (𝑥𝐴𝐵)
65a1i 11 . . . . . . . . 9 (𝜑𝐹 = (𝑥𝐴𝐵))
7 pimconstlt1.2 . . . . . . . . . 10 (𝜑𝐵 ∈ ℝ)
87adantr 473 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
96, 8fvmpt2d 6518 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
10 pimconstlt1.4 . . . . . . . . 9 (𝜑𝐵 < 𝐶)
1110adantr 473 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 < 𝐶)
129, 11eqbrtrd 4865 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) < 𝐶)
134, 12jca 508 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝐴 ∧ (𝐹𝑥) < 𝐶))
14 rabid 3297 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ↔ (𝑥𝐴 ∧ (𝐹𝑥) < 𝐶))
1513, 14sylibr 226 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
1615ex 402 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶}))
173, 16ralrimi 3138 . . 3 (𝜑 → ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
18 nfcv 2941 . . . 4 𝑥𝐴
19 nfrab1 3304 . . . 4 𝑥{𝑥𝐴 ∣ (𝐹𝑥) < 𝐶}
2018, 19dfss3f 3790 . . 3 (𝐴 ⊆ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ↔ ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
2117, 20sylibr 226 . 2 (𝜑𝐴 ⊆ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
222, 21eqssd 3815 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wnf 1879  wcel 2157  wral 3089  {crab 3093  wss 3769   class class class wbr 4843  cmpt 4922  cfv 6101  cr 10223   < clt 10363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fv 6109
This theorem is referenced by:  smfconst  41704
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