Theorem preimagelt 47982

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 2926	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfrab1 3435	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}
4	2, 3	nfdif 4069	. 2 ⊢ Ⅎ𝑥(𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
5		nfrab1 3435	. 2 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}
6		eldifi 4070	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) → 𝑥 ∈ 𝐴)
7	6	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝑥 ∈ 𝐴)
8		eldifn 4071	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) → ¬ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
9	6	anim1i 627	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ∧ 𝐶 ≤ 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝐵))
10		rabid 3436	. . . . . . . 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 828	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) → ¬ 𝐶 ≤ 𝐵)
13	12	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → ¬ 𝐶 ≤ 𝐵)
14		preimagelt.b	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
15	6, 14	sylan2 605	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝐵 ∈ ℝ*)
16		preimagelt.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
17	16	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝐶 ∈ ℝ*)
18	15, 17	xrltnled 11416	. . . . 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 3436	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 < 𝐶))
21	7, 19, 20	sylanbrc 595	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶})
22		rabidim1 3437	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} → 𝑥 ∈ 𝐴)
23	22	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝑥 ∈ 𝐴)
24		rabidim2 46390	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} → 𝐵 < 𝐶)
25	24	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝐵 < 𝐶)
26	22, 14	sylan2 605	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝐵 ∈ ℝ*)
27	16	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝐶 ∈ ℝ*)
28	26, 27	xrltnled 11416	. . . . . . 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 494	. . . . 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 4053	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}))
33	21, 32	impbida 813	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}))
34	1, 4, 5, 33	eqrd 3932	1 ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2147  {crab 3416   ∖ cdif 4048   class class class wbr 5108  ℝ^*cxr 11381   < clt 11382   ≤ cle 11383

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-cnv 5666  df-le 11388

This theorem is used by:  salpreimagelt  47990