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Theorem preimagelt 39589
 Description: The preimage of a right-open, unbounded below interval, is the complement of a left-close, unbounded above interval. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
preimagelt.x 𝑥𝜑
preimagelt.b ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
preimagelt.c (𝜑𝐶 ∈ ℝ*)
Assertion
Ref Expression
preimagelt (𝜑 → (𝐴 ∖ {𝑥𝐴𝐶𝐵}) = {𝑥𝐴𝐵 < 𝐶})
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem preimagelt
StepHypRef Expression
1 preimagelt.x . . 3 𝑥𝜑
2 eldifi 3694 . . . . . . . 8 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → 𝑥𝐴)
32adantl 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝑥𝐴)
42anim1i 590 . . . . . . . . . . 11 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ∧ 𝐶𝐵) → (𝑥𝐴𝐶𝐵))
5 rabid 3095 . . . . . . . . . . 11 (𝑥 ∈ {𝑥𝐴𝐶𝐵} ↔ (𝑥𝐴𝐶𝐵))
64, 5sylibr 223 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ∧ 𝐶𝐵) → 𝑥 ∈ {𝑥𝐴𝐶𝐵})
7 eldifn 3695 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → ¬ 𝑥 ∈ {𝑥𝐴𝐶𝐵})
87adantr 480 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ∧ 𝐶𝐵) → ¬ 𝑥 ∈ {𝑥𝐴𝐶𝐵})
96, 8pm2.65da 598 . . . . . . . . 9 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → ¬ 𝐶𝐵)
109adantl 481 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → ¬ 𝐶𝐵)
11 preimagelt.b . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
123, 11syldan 486 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝐵 ∈ ℝ*)
13 preimagelt.c . . . . . . . . . 10 (𝜑𝐶 ∈ ℝ*)
1413adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝐶 ∈ ℝ*)
1512, 14xrltnled 38520 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
1610, 15mpbird 246 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝐵 < 𝐶)
173, 16jca 553 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → (𝑥𝐴𝐵 < 𝐶))
18 rabid 3095 . . . . . 6 (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} ↔ (𝑥𝐴𝐵 < 𝐶))
1917, 18sylibr 223 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶})
2019ex 449 . . . 4 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}))
2118simplbi 475 . . . . . . 7 (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} → 𝑥𝐴)
2221adantl 481 . . . . . 6 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝑥𝐴)
2318simprbi 479 . . . . . . . . . 10 (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} → 𝐵 < 𝐶)
2423adantl 481 . . . . . . . . 9 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝐵 < 𝐶)
2522, 11syldan 486 . . . . . . . . . 10 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝐵 ∈ ℝ*)
2613adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝐶 ∈ ℝ*)
2725, 26xrltnled 38520 . . . . . . . . 9 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
2824, 27mpbid 221 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → ¬ 𝐶𝐵)
2928intnand 953 . . . . . . 7 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → ¬ (𝑥𝐴𝐶𝐵))
3029, 5sylnibr 318 . . . . . 6 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → ¬ 𝑥 ∈ {𝑥𝐴𝐶𝐵})
3122, 30eldifd 3551 . . . . 5 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}))
3231ex 449 . . . 4 (𝜑 → (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} → 𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})))
3320, 32impbid 201 . . 3 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ↔ 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}))
341, 33alrimi 2069 . 2 (𝜑 → ∀𝑥(𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ↔ 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}))
35 nfcv 2751 . . . 4 𝑥𝐴
36 nfrab1 3099 . . . 4 𝑥{𝑥𝐴𝐶𝐵}
3735, 36nfdif 3693 . . 3 𝑥(𝐴 ∖ {𝑥𝐴𝐶𝐵})
38 nfrab1 3099 . . 3 𝑥{𝑥𝐴𝐵 < 𝐶}
3937, 38dfcleqf 38281 . 2 ((𝐴 ∖ {𝑥𝐴𝐶𝐵}) = {𝑥𝐴𝐵 < 𝐶} ↔ ∀𝑥(𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ↔ 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}))
4034, 39sylibr 223 1 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐶𝐵}) = {𝑥𝐴𝐵 < 𝐶})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  {crab 2900   ∖ cdif 3537   class class class wbr 4583  ℝ*cxr 9952   < clt 9953   ≤ cle 9954 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-le 9959 This theorem is referenced by:  salpreimagelt  39595
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