Theorem preimagelt 41427

Description: The preimage of a right-open, unbounded below interval, is the complement of a left-close, 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	. . 3 ⊢ Ⅎ𝑥𝜑
2		eldifi 3883	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) → 𝑥 ∈ 𝐴)
3	2	adantl 467	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝑥 ∈ 𝐴)
4	2	anim1i 602	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ∧ 𝐶 ≤ 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝐵))
5		rabid 3264	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝐵))
6	4, 5	sylibr 224	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ∧ 𝐶 ≤ 𝐵) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
7		eldifn 3884	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) → ¬ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
8	7	adantr 466	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ∧ 𝐶 ≤ 𝐵) → ¬ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
9	6, 8	pm2.65da 818	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) → ¬ 𝐶 ≤ 𝐵)
10	9	adantl 467	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → ¬ 𝐶 ≤ 𝐵)
11		preimagelt.b	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)
12	3, 11	syldan 579	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝐵 ∈ ℝ*)
13		preimagelt.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ*)
14	13	adantr 466	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝐶 ∈ ℝ*)
15	12, 14	xrltnled 40090	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
16	10, 15	mpbird 247	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝐵 < 𝐶)
17	3, 16	jca 501	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → (𝑥 ∈ 𝐴 ∧ 𝐵 < 𝐶))
18		rabid 3264	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 < 𝐶))
19	17, 18	sylibr 224	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶})
20	19	ex 397	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}))
21	18	simplbi 485	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} → 𝑥 ∈ 𝐴)
22	21	adantl 467	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝑥 ∈ 𝐴)
23	18	simprbi 484	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} → 𝐵 < 𝐶)
24	23	adantl 467	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝐵 < 𝐶)
25	22, 11	syldan 579	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝐵 ∈ ℝ*)
26	13	adantr 466	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝐶 ∈ ℝ*)
27	25, 26	xrltnled 40090	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
28	24, 27	mpbid 222	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → ¬ 𝐶 ≤ 𝐵)
29	28	intnand 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → ¬ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝐵))
30	29, 5	sylnibr 318	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → ¬ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
31	22, 30	eldifd 3734	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) → 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}))
32	31	ex 397	. . . 4 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} → 𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})))
33	20, 32	impbid 202	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}))
34	1, 33	alrimi 2238	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}))
35		nfcv 2913	. . . 4 ⊢ Ⅎ𝑥𝐴
36		nfrab1 3271	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}
37	35, 36	nfdif 3882	. . 3 ⊢ Ⅎ𝑥(𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})
38		nfrab1 3271	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}
39	37, 38	dfcleqf 39774	. 2 ⊢ ((𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ↔ ∀𝑥(𝑥 ∈ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}))
40	34, 39	sylibr 224	1 ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382  ∀wal 1629   = wceq 1631  Ⅎwnf 1856   ∈ wcel 2145  {crab 3065   ∖ cdif 3720   class class class wbr 4787  ℝ^*cxr 10279   < clt 10280   ≤ cle 10281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-le 10286

This theorem is referenced by:  salpreimagelt  41433