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Theorem pimconstlt1 39375
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 3649 . . 3 {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ⊆ 𝐴
21a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ⊆ 𝐴)
3 pimconstlt1.1 . . . 4 𝑥𝜑
4 simpr 475 . . . . . . 7 ((𝜑𝑥𝐴) → 𝑥𝐴)
5 pimconstlt1.3 . . . . . . . . . 10 𝐹 = (𝑥𝐴𝐵)
65a1i 11 . . . . . . . . 9 (𝜑𝐹 = (𝑥𝐴𝐵))
7 pimconstlt1.2 . . . . . . . . . 10 (𝜑𝐵 ∈ ℝ)
87adantr 479 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
96, 8fvmpt2d 6186 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
10 pimconstlt1.4 . . . . . . . . 9 (𝜑𝐵 < 𝐶)
1110adantr 479 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 < 𝐶)
129, 11eqbrtrd 4599 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) < 𝐶)
134, 12jca 552 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝐴 ∧ (𝐹𝑥) < 𝐶))
14 rabid 3094 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ↔ (𝑥𝐴 ∧ (𝐹𝑥) < 𝐶))
1513, 14sylibr 222 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
1615ex 448 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶}))
173, 16ralrimi 2939 . . 3 (𝜑 → ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
18 nfcv 2750 . . . 4 𝑥𝐴
19 nfrab1 3098 . . . 4 𝑥{𝑥𝐴 ∣ (𝐹𝑥) < 𝐶}
2018, 19dfss3f 3559 . . 3 (𝐴 ⊆ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ↔ ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
2117, 20sylibr 222 . 2 (𝜑𝐴 ⊆ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
222, 21eqssd 3584 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wnf 1698  wcel 1976  wral 2895  {crab 2899  wss 3539   class class class wbr 4577  cmpt 4637  cfv 5789  cr 9791   < clt 9930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fv 5797
This theorem is referenced by:  smfconst  39419
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