Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pimconstlt1 Structured version   Visualization version   GIF version

Theorem pimconstlt1 41236
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 3720 . . 3 {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ⊆ 𝐴
21a1i 11 . 2 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ⊆ 𝐴)
3 pimconstlt1.1 . . . 4 𝑥𝜑
4 simpr 476 . . . . . . 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 6332 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
10 pimconstlt1.4 . . . . . . . . 9 (𝜑𝐵 < 𝐶)
1110adantr 480 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 < 𝐶)
129, 11eqbrtrd 4707 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) < 𝐶)
134, 12jca 553 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝐴 ∧ (𝐹𝑥) < 𝐶))
14 rabid 3145 . . . . . 6 (𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ↔ (𝑥𝐴 ∧ (𝐹𝑥) < 𝐶))
1513, 14sylibr 224 . . . . 5 ((𝜑𝑥𝐴) → 𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
1615ex 449 . . . 4 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶}))
173, 16ralrimi 2986 . . 3 (𝜑 → ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
18 nfcv 2793 . . . 4 𝑥𝐴
19 nfrab1 3152 . . . 4 𝑥{𝑥𝐴 ∣ (𝐹𝑥) < 𝐶}
2018, 19dfss3f 3628 . . 3 (𝐴 ⊆ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} ↔ ∀𝑥𝐴 𝑥 ∈ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
2117, 20sylibr 224 . 2 (𝜑𝐴 ⊆ {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶})
222, 21eqssd 3653 1 (𝜑 → {𝑥𝐴 ∣ (𝐹𝑥) < 𝐶} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wnf 1748  wcel 2030  wral 2941  {crab 2945  wss 3607   class class class wbr 4685  cmpt 4762  cfv 5926  cr 9973   < clt 10112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by:  smfconst  41279
  Copyright terms: Public domain W3C validator