Theorem restuni3 45123

Description: The underlying set of a subspace induced by the subspace operator ↾_t. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
restuni3.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
restuni3.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
restuni3	⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵))

Proof of Theorem restuni3

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni2 4911	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ (𝐴 ↾t 𝐵) ↔ ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
2	1	biimpi 216	. . . . . . 7 ⊢ (𝑥 ∈ ∪ (𝐴 ↾t 𝐵) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
3	2	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
4		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵)) → 𝑦 ∈ (𝐴 ↾t 𝐵))
5		restuni3.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		restuni3.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		elrest 17472	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑦 ∈ (𝐴 ↾t 𝐵) ↔ ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵)))
8	5, 6, 7	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑦 ∈ (𝐴 ↾t 𝐵) ↔ ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵)))
9	8	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵)) → (𝑦 ∈ (𝐴 ↾t 𝐵) ↔ ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵)))
10	4, 9	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵)) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵))
11	10	3adant3 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ 𝑦) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵))
12		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐵)) → 𝑥 ∈ 𝑦)
13		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐵)) → 𝑦 = (𝑧 ∩ 𝐵))
14	12, 13	eleqtrd 2843	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐵)) → 𝑥 ∈ (𝑧 ∩ 𝐵))
15	14	ex 412	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑦 → (𝑦 = (𝑧 ∩ 𝐵) → 𝑥 ∈ (𝑧 ∩ 𝐵)))
16	15	3ad2ant3 1136	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ 𝑦) → (𝑦 = (𝑧 ∩ 𝐵) → 𝑥 ∈ (𝑧 ∩ 𝐵)))
17	16	reximdv 3170	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ 𝑦) → (∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵) → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵)))
18	11, 17	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ 𝑦) → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵))
19	18	3exp 1120	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐴 ↾t 𝐵) → (𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵))))
20	19	rexlimdv 3153	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵)))
21	20	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) → (∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵)))
22	3, 21	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵))
23		elinel1 4201	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ 𝑧)
24	23	adantl 481	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ 𝑧)
25		simpl 482	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑧 ∈ 𝐴)
26		elunii 4912	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → 𝑥 ∈ ∪ 𝐴)
27	24, 25, 26	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ ∪ 𝐴)
28		elinel2 4202	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ 𝐵)
29	28	adantl 481	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ 𝐵)
30	27, 29	elind 4200	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
31	30	ex 412	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → (𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)))
32	31	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) ∧ 𝑧 ∈ 𝐴) → (𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)))
33	32	rexlimdva 3155	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) → (∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)))
34	22, 33	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
35	34	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ∪ (𝐴 ↾t 𝐵)𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
36		dfss3 3972	. . 3 ⊢ (∪ (𝐴 ↾t 𝐵) ⊆ (∪ 𝐴 ∩ 𝐵) ↔ ∀𝑥 ∈ ∪ (𝐴 ↾t 𝐵)𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
37	35, 36	sylibr 234	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) ⊆ (∪ 𝐴 ∩ 𝐵))
38		elinel1 4201	. . . . . 6 ⊢ (𝑥 ∈ (∪ 𝐴 ∩ 𝐵) → 𝑥 ∈ ∪ 𝐴)
39		eluni2 4911	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧)
40	38, 39	sylib 218	. . . . 5 ⊢ (𝑥 ∈ (∪ 𝐴 ∩ 𝐵) → ∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧)
41	40	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → ∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧)
42	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐴 ∈ 𝑉)
43	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐵 ∈ 𝑊)
44		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
45		eqid 2737	. . . . . . . . . 10 ⊢ (𝑧 ∩ 𝐵) = (𝑧 ∩ 𝐵)
46	42, 43, 44, 45	elrestd 45113	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵))
47	46	3adant3 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → (𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵))
48	47	3adant1r 1178	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → (𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵))
49		simp3 1139	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝑧)
50		simp1r 1199	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
51		elinel2 4202	. . . . . . . . 9 ⊢ (𝑥 ∈ (∪ 𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐵)
52	50, 51	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝐵)
53		simpl 482	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝑧)
54		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
55	53, 54	elind 4200	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝑧 ∩ 𝐵))
56	49, 52, 55	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ (𝑧 ∩ 𝐵))
57		eleq2 2830	. . . . . . . 8 ⊢ (𝑦 = (𝑧 ∩ 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑧 ∩ 𝐵)))
58	57	rspcev 3622	. . . . . . 7 ⊢ (((𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
59	48, 56, 58	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
60	59	3exp 1120	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → (𝑧 ∈ 𝐴 → (𝑥 ∈ 𝑧 → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)))
61	60	rexlimdv 3153	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → (∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧 → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦))
62	41, 61	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
63	62, 1	sylibr 234	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → 𝑥 ∈ ∪ (𝐴 ↾t 𝐵))
64	37, 63	eqelssd 4005	1 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ∩ cin 3950   ⊆ wss 3951  ∪ cuni 4907  (class class class)co 7431   ↾_t crest 17465

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rest 17467

This theorem is referenced by:  restuni4  45126  subsalsal  46374