Theorem preimagelt 44467

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	⊢ Ⅎ𝑥𝜑
preimagelt.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
preimagelt.c	⊢ (𝜑 → 𝐶 ∈ ℝ*)

Assertion

Ref	Expression
preimagelt	⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶})

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem preimagelt

Step	Hyp	Ref	Expression
1		preimagelt.x	. 2 ⊢ Ⅎ𝑥𝜑
2		nfcv 2905	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfrab1 3336	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}
4	2, 3	nfdif 4066	. 2 ⊢ Ⅎ𝑥(𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
5		nfrab1 3336	. 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}
6		eldifi 4067	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) → 𝑥 ∈ 𝐴)
7	6	adantl 483	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝑥 ∈ 𝐴)
8		eldifn 4068	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) → ¬ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
9	6	anim1i 616	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ∧ 𝐶 ≤ 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝐵))
10		rabid 3329	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝐵))
11	9, 10	sylibr 233	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ∧ 𝐶 ≤ 𝐵) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
12	8, 11	mtand 814	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) → ¬ 𝐶 ≤ 𝐵)
13	12	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → ¬ 𝐶 ≤ 𝐵)
14		preimagelt.b	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
15	6, 14	sylan2 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝐵 ∈ ℝ*)
16		preimagelt.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
17	16	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝐶 ∈ ℝ*)
18	15, 17	xrltnled 43130	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
19	13, 18	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝐵 < 𝐶)
20		rabid 3329	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 < 𝐶))
21	7, 19, 20	sylanbrc 584	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶})
22		rabidim1 3331	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} → 𝑥 ∈ 𝐴)
23	22	adantl 483	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝑥 ∈ 𝐴)
24		rabidim2 42865	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} → 𝐵 < 𝐶)
25	24	adantl 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝐵 < 𝐶)
26	22, 14	sylan2 594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝐵 ∈ ℝ*)
27	16	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝐶 ∈ ℝ*)
28	26, 27	xrltnled 43130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
29	25, 28	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → ¬ 𝐶 ≤ 𝐵)
30	29	intnand 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → ¬ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝐵))
31	30, 10	sylnibr 329	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → ¬ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
32	23, 31	eldifd 3903	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}))
33	21, 32	impbida 799	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}))
34	1, 4, 5, 33	eqrd 3945	1 ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1539  Ⅎwnf 1783   ∈ wcel 2104  {crab 3303   ∖ cdif 3889   class class class wbr 5081  ℝ^*cxr 11058   < clt 11059   ≤ cle 11060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-xp 5606  df-cnv 5608  df-le 11065

This theorem is referenced by:  salpreimagelt  44475