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Theorem preimagelt 41418
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 3875 . . . . . . . 8 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → 𝑥𝐴)
32adantl 473 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝑥𝐴)
42anim1i 593 . . . . . . . . . . 11 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ∧ 𝐶𝐵) → (𝑥𝐴𝐶𝐵))
5 rabid 3254 . . . . . . . . . . 11 (𝑥 ∈ {𝑥𝐴𝐶𝐵} ↔ (𝑥𝐴𝐶𝐵))
64, 5sylibr 224 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ∧ 𝐶𝐵) → 𝑥 ∈ {𝑥𝐴𝐶𝐵})
7 eldifn 3876 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → ¬ 𝑥 ∈ {𝑥𝐴𝐶𝐵})
87adantr 472 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ∧ 𝐶𝐵) → ¬ 𝑥 ∈ {𝑥𝐴𝐶𝐵})
96, 8pm2.65da 601 . . . . . . . . 9 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → ¬ 𝐶𝐵)
109adantl 473 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → ¬ 𝐶𝐵)
11 preimagelt.b . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
123, 11syldan 488 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝐵 ∈ ℝ*)
13 preimagelt.c . . . . . . . . . 10 (𝜑𝐶 ∈ ℝ*)
1413adantr 472 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝐶 ∈ ℝ*)
1512, 14xrltnled 40077 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
1610, 15mpbird 247 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝐵 < 𝐶)
173, 16jca 555 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → (𝑥𝐴𝐵 < 𝐶))
18 rabid 3254 . . . . . 6 (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} ↔ (𝑥𝐴𝐵 < 𝐶))
1917, 18sylibr 224 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶})
2019ex 449 . . . 4 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}))
2118simplbi 478 . . . . . . 7 (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} → 𝑥𝐴)
2221adantl 473 . . . . . 6 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝑥𝐴)
2318simprbi 483 . . . . . . . . . 10 (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} → 𝐵 < 𝐶)
2423adantl 473 . . . . . . . . 9 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝐵 < 𝐶)
2522, 11syldan 488 . . . . . . . . . 10 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝐵 ∈ ℝ*)
2613adantr 472 . . . . . . . . . 10 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝐶 ∈ ℝ*)
2725, 26xrltnled 40077 . . . . . . . . 9 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
2824, 27mpbid 222 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → ¬ 𝐶𝐵)
2928intnand 1000 . . . . . . 7 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → ¬ (𝑥𝐴𝐶𝐵))
3029, 5sylnibr 318 . . . . . 6 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → ¬ 𝑥 ∈ {𝑥𝐴𝐶𝐵})
3122, 30eldifd 3726 . . . . 5 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}))
3231ex 449 . . . 4 (𝜑 → (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} → 𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})))
3320, 32impbid 202 . . 3 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ↔ 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}))
341, 33alrimi 2229 . 2 (𝜑 → ∀𝑥(𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ↔ 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}))
35 nfcv 2902 . . . 4 𝑥𝐴
36 nfrab1 3261 . . . 4 𝑥{𝑥𝐴𝐶𝐵}
3735, 36nfdif 3874 . . 3 𝑥(𝐴 ∖ {𝑥𝐴𝐶𝐵})
38 nfrab1 3261 . . 3 𝑥{𝑥𝐴𝐵 < 𝐶}
3937, 38dfcleqf 39754 . 2 ((𝐴 ∖ {𝑥𝐴𝐶𝐵}) = {𝑥𝐴𝐵 < 𝐶} ↔ ∀𝑥(𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ↔ 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}))
4034, 39sylibr 224 1 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐶𝐵}) = {𝑥𝐴𝐵 < 𝐶})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wnf 1857  wcel 2139  {crab 3054  cdif 3712   class class class wbr 4804  *cxr 10265   < clt 10266  cle 10267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-cnv 5274  df-le 10272
This theorem is referenced by:  salpreimagelt  41424
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