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

Theorem preimagelt 46655
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 2903 . . 3 𝑥𝐴
3 nfrab1 3454 . . 3 𝑥{𝑥𝐴𝐶𝐵}
42, 3nfdif 4139 . 2 𝑥(𝐴 ∖ {𝑥𝐴𝐶𝐵})
5 nfrab1 3454 . 2 𝑥{𝑥𝐴𝐵 < 𝐶}
6 eldifi 4141 . . . . 5 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → 𝑥𝐴)
76adantl 481 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝑥𝐴)
8 eldifn 4142 . . . . . . 7 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → ¬ 𝑥 ∈ {𝑥𝐴𝐶𝐵})
96anim1i 615 . . . . . . . 8 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ∧ 𝐶𝐵) → (𝑥𝐴𝐶𝐵))
10 rabid 3455 . . . . . . . 8 (𝑥 ∈ {𝑥𝐴𝐶𝐵} ↔ (𝑥𝐴𝐶𝐵))
119, 10sylibr 234 . . . . . . 7 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ∧ 𝐶𝐵) → 𝑥 ∈ {𝑥𝐴𝐶𝐵})
128, 11mtand 816 . . . . . 6 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → ¬ 𝐶𝐵)
1312adantl 481 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → ¬ 𝐶𝐵)
14 preimagelt.b . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
156, 14sylan2 593 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝐵 ∈ ℝ*)
16 preimagelt.c . . . . . . 7 (𝜑𝐶 ∈ ℝ*)
1716adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝐶 ∈ ℝ*)
1815, 17xrltnled 45313 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
1913, 18mpbird 257 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝐵 < 𝐶)
20 rabid 3455 . . . 4 (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} ↔ (𝑥𝐴𝐵 < 𝐶))
217, 19, 20sylanbrc 583 . . 3 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶})
22 rabidim1 3456 . . . . 5 (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} → 𝑥𝐴)
2322adantl 481 . . . 4 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝑥𝐴)
24 rabidim2 45042 . . . . . . . 8 (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} → 𝐵 < 𝐶)
2524adantl 481 . . . . . . 7 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝐵 < 𝐶)
2622, 14sylan2 593 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝐵 ∈ ℝ*)
2716adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝐶 ∈ ℝ*)
2826, 27xrltnled 45313 . . . . . . 7 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
2925, 28mpbid 232 . . . . . 6 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → ¬ 𝐶𝐵)
3029intnand 488 . . . . 5 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → ¬ (𝑥𝐴𝐶𝐵))
3130, 10sylnibr 329 . . . 4 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → ¬ 𝑥 ∈ {𝑥𝐴𝐶𝐵})
3223, 31eldifd 3974 . . 3 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}))
3321, 32impbida 801 . 2 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ↔ 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}))
341, 4, 5, 33eqrd 4015 1 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐶𝐵}) = {𝑥𝐴𝐵 < 𝐶})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wnf 1780  wcel 2106  {crab 3433  cdif 3960   class class class wbr 5148  *cxr 11292   < clt 11293  cle 11294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-le 11299
This theorem is referenced by:  salpreimagelt  46663
  Copyright terms: Public domain W3C validator