Theorem preimagelt 47304

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 2931	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfrab1 3443	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}
4	2, 3	nfdif 4092	. 2 ⊢ Ⅎ𝑥(𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
5		nfrab1 3443	. 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}
6		eldifi 4093	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) → 𝑥 ∈ 𝐴)
7	6	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝑥 ∈ 𝐴)
8		eldifn 4094	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) → ¬ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
9	6	anim1i 626	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ∧ 𝐶 ≤ 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝐵))
10		rabid 3444	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝐵))
11	9, 10	sylibr 237	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ∧ 𝐶 ≤ 𝐵) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
12	8, 11	mtand 827	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) → ¬ 𝐶 ≤ 𝐵)
13	12	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → ¬ 𝐶 ≤ 𝐵)
14		preimagelt.b	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
15	6, 14	sylan2 604	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝐵 ∈ ℝ*)
16		preimagelt.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
17	16	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝐶 ∈ ℝ*)
18	15, 17	xrltnled 11276	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
19	13, 18	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝐵 < 𝐶)
20		rabid 3444	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 < 𝐶))
21	7, 19, 20	sylanbrc 594	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶})
22		rabidim1 3445	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} → 𝑥 ∈ 𝐴)
23	22	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝑥 ∈ 𝐴)
24		rabidim2 45711	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} → 𝐵 < 𝐶)
25	24	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝐵 < 𝐶)
26	22, 14	sylan2 604	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝐵 ∈ ℝ*)
27	16	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝐶 ∈ ℝ*)
28	26, 27	xrltnled 11276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
29	25, 28	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → ¬ 𝐶 ≤ 𝐵)
30	29	intnand 493	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → ¬ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝐵))
31	30, 10	sylnibr 332	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → ¬ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
32	23, 31	eldifd 3924	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}))
33	21, 32	impbida 812	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}))
34	1, 4, 5, 33	eqrd 3964	1 ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567  Ⅎwnf 1810   ∈ wcel 2149  {crab 3423   ∖ cdif 3910   class class class wbr 5113  ℝ^*cxr 11241   < clt 11242   ≤ cle 11243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-cnv 5670  df-le 11248

This theorem is referenced by:  salpreimagelt  47312