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Theorem preimagelt 45415
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 2904 . . 3 𝑥𝐴
3 nfrab1 3452 . . 3 𝑥{𝑥𝐴𝐶𝐵}
42, 3nfdif 4126 . 2 𝑥(𝐴 ∖ {𝑥𝐴𝐶𝐵})
5 nfrab1 3452 . 2 𝑥{𝑥𝐴𝐵 < 𝐶}
6 eldifi 4127 . . . . 5 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → 𝑥𝐴)
76adantl 483 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝑥𝐴)
8 eldifn 4128 . . . . . . 7 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → ¬ 𝑥 ∈ {𝑥𝐴𝐶𝐵})
96anim1i 616 . . . . . . . 8 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ∧ 𝐶𝐵) → (𝑥𝐴𝐶𝐵))
10 rabid 3453 . . . . . . . 8 (𝑥 ∈ {𝑥𝐴𝐶𝐵} ↔ (𝑥𝐴𝐶𝐵))
119, 10sylibr 233 . . . . . . 7 ((𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ∧ 𝐶𝐵) → 𝑥 ∈ {𝑥𝐴𝐶𝐵})
128, 11mtand 815 . . . . . 6 (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) → ¬ 𝐶𝐵)
1312adantl 483 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → ¬ 𝐶𝐵)
14 preimagelt.b . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ*)
156, 14sylan2 594 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝐵 ∈ ℝ*)
16 preimagelt.c . . . . . . 7 (𝜑𝐶 ∈ ℝ*)
1716adantr 482 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝐶 ∈ ℝ*)
1815, 17xrltnled 44073 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
1913, 18mpbird 257 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝐵 < 𝐶)
20 rabid 3453 . . . 4 (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} ↔ (𝑥𝐴𝐵 < 𝐶))
217, 19, 20sylanbrc 584 . . 3 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵})) → 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶})
22 rabidim1 3454 . . . . 5 (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} → 𝑥𝐴)
2322adantl 483 . . . 4 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝑥𝐴)
24 rabidim2 43791 . . . . . . . 8 (𝑥 ∈ {𝑥𝐴𝐵 < 𝐶} → 𝐵 < 𝐶)
2524adantl 483 . . . . . . 7 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝐵 < 𝐶)
2622, 14sylan2 594 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝐵 ∈ ℝ*)
2716adantr 482 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝐶 ∈ ℝ*)
2826, 27xrltnled 44073 . . . . . . 7 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → (𝐵 < 𝐶 ↔ ¬ 𝐶𝐵))
2925, 28mpbid 231 . . . . . 6 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → ¬ 𝐶𝐵)
3029intnand 490 . . . . 5 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → ¬ (𝑥𝐴𝐶𝐵))
3130, 10sylnibr 329 . . . 4 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → ¬ 𝑥 ∈ {𝑥𝐴𝐶𝐵})
3223, 31eldifd 3960 . . 3 ((𝜑𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}) → 𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}))
3321, 32impbida 800 . 2 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥𝐴𝐶𝐵}) ↔ 𝑥 ∈ {𝑥𝐴𝐵 < 𝐶}))
341, 4, 5, 33eqrd 4002 1 (𝜑 → (𝐴 ∖ {𝑥𝐴𝐶𝐵}) = {𝑥𝐴𝐵 < 𝐶})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wnf 1786  wcel 2107  {crab 3433  cdif 3946   class class class wbr 5149  *cxr 11247   < clt 11248  cle 11249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-le 11254
This theorem is referenced by:  salpreimagelt  45423
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