Theorem restuni3 38333

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 4376	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ (𝐴 ↾t 𝐵) ↔ ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
2	1	biimpi 205	. . . . . . 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 476	. . . . . . . . . . . 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 15911	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑦 ∈ (𝐴 ↾t 𝐵) ↔ ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵)))
8	5, 6, 7	syl2anc 691	. . . . . . . . . . . . 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 221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵)) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵))
11	10	3adant3 1074	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ 𝑦) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵))
12		simpl 472	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐵)) → 𝑥 ∈ 𝑦)
13		simpr 476	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐵)) → 𝑦 = (𝑧 ∩ 𝐵))
14	12, 13	eleqtrd 2690	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐵)) → 𝑥 ∈ (𝑧 ∩ 𝐵))
15	14	ex 449	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑦 → (𝑦 = (𝑧 ∩ 𝐵) → 𝑥 ∈ (𝑧 ∩ 𝐵)))
16	15	3ad2ant3 1077	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ 𝑦) → (𝑦 = (𝑧 ∩ 𝐵) → 𝑥 ∈ (𝑧 ∩ 𝐵)))
17	16	reximdv 2999	. . . . . . . . . 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 1256	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐴 ↾t 𝐵) → (𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵))))
20	19	rexlimdv 3012	. . . . . . 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 3761	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ 𝑧)
24	23	adantl 481	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ 𝑧)
25		simpl 472	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑧 ∈ 𝐴)
26		elunii 4377	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → 𝑥 ∈ ∪ 𝐴)
27	24, 25, 26	syl2anc 691	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ ∪ 𝐴)
28		elinel2 3762	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ 𝐵)
29	28	adantl 481	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ 𝐵)
30	27, 29	elind 3760	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
31	30	ex 449	. . . . . . 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 3013	. . . . 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 2949	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ∪ (𝐴 ↾t 𝐵)𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
36		dfss3 3558	. . 3 ⊢ (∪ (𝐴 ↾t 𝐵) ⊆ (∪ 𝐴 ∩ 𝐵) ↔ ∀𝑥 ∈ ∪ (𝐴 ↾t 𝐵)𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
37	35, 36	sylibr 223	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) ⊆ (∪ 𝐴 ∩ 𝐵))
38		elinel1 3761	. . . . . . . 8 ⊢ (𝑥 ∈ (∪ 𝐴 ∩ 𝐵) → 𝑥 ∈ ∪ 𝐴)
39		eluni2 4376	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧)
40	38, 39	sylib 207	. . . . . . 7 ⊢ (𝑥 ∈ (∪ 𝐴 ∩ 𝐵) → ∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧)
41	40	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → ∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧)
42	5	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐴 ∈ 𝑉)
43	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐵 ∈ 𝑊)
44		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
45		eqid 2610	. . . . . . . . . . . 12 ⊢ (𝑧 ∩ 𝐵) = (𝑧 ∩ 𝐵)
46	42, 43, 44, 45	elrestd 38322	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵))
47	46	3adant3 1074	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → (𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵))
48	47	3adant1r 1311	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → (𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵))
49		simp3 1056	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝑧)
50		simp1r 1079	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
51		elinel2 3762	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (∪ 𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐵)
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝐵)
53		simpl 472	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝑧)
54		simpr 476	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
55	53, 54	elind 3760	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝑧 ∩ 𝐵))
56	49, 52, 55	syl2anc 691	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ (𝑧 ∩ 𝐵))
57		eleq2 2677	. . . . . . . . . 10 ⊢ (𝑦 = (𝑧 ∩ 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑧 ∩ 𝐵)))
58	57	rspcev 3282	. . . . . . . . 9 ⊢ (((𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
59	48, 56, 58	syl2anc 691	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
60	59	3exp 1256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → (𝑧 ∈ 𝐴 → (𝑥 ∈ 𝑧 → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)))
61	60	rexlimdv 3012	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → (∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧 → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦))
62	41, 61	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
63	62, 1	sylibr 223	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → 𝑥 ∈ ∪ (𝐴 ↾t 𝐵))
64	63	ralrimiva 2949	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (∪ 𝐴 ∩ 𝐵)𝑥 ∈ ∪ (𝐴 ↾t 𝐵))
65		dfss3 3558	. . 3 ⊢ ((∪ 𝐴 ∩ 𝐵) ⊆ ∪ (𝐴 ↾t 𝐵) ↔ ∀𝑥 ∈ (∪ 𝐴 ∩ 𝐵)𝑥 ∈ ∪ (𝐴 ↾t 𝐵))
66	64, 65	sylibr 223	. 2 ⊢ (𝜑 → (∪ 𝐴 ∩ 𝐵) ⊆ ∪ (𝐴 ↾t 𝐵))
67	37, 66	eqssd 3585	1 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  ∪ cuni 4372  (class class class)co 6549   ↾_t crest 15904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-rest 15906

This theorem is referenced by:  restuni4  38336  subsalsal  39253