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Theorem preimagelt 43334
Description: The preimage of a right-open, unbounded below interval, is the complement of a left-closed 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 . 2 𝑥𝜑
2 nfcv 2958 . . 3 𝑥𝐴
3 nfrab1 3340 . . 3 𝑥{𝑥𝐴𝐶𝐵}
42, 3nfdif 4056 . 2 𝑥(𝐴 ∖ {𝑥𝐴𝐶𝐵})
5 nfrab1 3340 . 2 𝑥{𝑥𝐴𝐵 < 𝐶}
6 eldifi 4057 . . . . 5 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → 𝑥𝐴)
76adantl 485 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝑥𝐴)
8 eldifn 4058 . . . . . . 7 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → ¬ 𝑥 ∈ {𝑥𝐴𝐶𝐵})
96anim1i 617 . . . . . . . 8 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ∧ 𝐶𝐵) → (𝑥𝐴𝐶𝐵))
10 rabid 3334 . . . . . . . 8 (𝑥 ∈ {𝑥𝐴𝐶𝐵} ↔ (𝑥𝐴𝐶𝐵))
119, 10sylibr 237 . . . . . . 7 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ∧ 𝐶𝐵) → 𝑥 ∈ {𝑥𝐴𝐶𝐵})
128, 11mtand 815 . . . . . 6 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → ¬ 𝐶𝐵)
1312adantl 485 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → ¬ 𝐶𝐵)
14 preimagelt.b . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
156, 14sylan2 595 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝐵 ∈ ℝ*)
16 preimagelt.c . . . . . . 7 (𝜑𝐶 ∈ ℝ*)
1716adantr 484 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝐶 ∈ ℝ*)
1815, 17xrltnled 41992 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
1913, 18mpbird 260 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝐵 < 𝐶)
20 rabid 3334 . . . 4 (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} ↔ (𝑥𝐴𝐵 < 𝐶))
217, 19, 20sylanbrc 586 . . 3 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶})
22 rabidim1 3336 . . . . 5 (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} → 𝑥𝐴)
2322adantl 485 . . . 4 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝑥𝐴)
24 rabidim2 41735 . . . . . . . 8 (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} → 𝐵 < 𝐶)
2524adantl 485 . . . . . . 7 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝐵 < 𝐶)
2622, 14sylan2 595 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝐵 ∈ ℝ*)
2716adantr 484 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝐶 ∈ ℝ*)
2826, 27xrltnled 41992 . . . . . . 7 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
2925, 28mpbid 235 . . . . . 6 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → ¬ 𝐶𝐵)
3029intnand 492 . . . . 5 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → ¬ (𝑥𝐴𝐶𝐵))
3130, 10sylnibr 332 . . . 4 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → ¬ 𝑥 ∈ {𝑥𝐴𝐶𝐵})
3223, 31eldifd 3895 . . 3 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}))
3321, 32impbida 800 . 2 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ↔ 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}))
341, 4, 5, 33eqrd 3937 1 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐶𝐵}) = {𝑥𝐴𝐵 < 𝐶})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wnf 1785  wcel 2112  {crab 3113  cdif 3881   class class class wbr 5033  *cxr 10667   < clt 10668  cle 10669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-cnv 5531  df-le 10674
This theorem is referenced by:  salpreimagelt  43340
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