Theorem restuni3 45816

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 4877	. . . . . . 7 ⊢ (𝑥 ∈ ∪ (𝐴 ↾t 𝐵) ↔ ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
2	1	bilani 509	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
3		simpr 489	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵)) → 𝑦 ∈ (𝐴 ↾t 𝐵))
4		restuni3.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		restuni3.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
6		elrest 17481	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑦 ∈ (𝐴 ↾t 𝐵) ↔ ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵)))
7	4, 5, 6	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑦 ∈ (𝐴 ↾t 𝐵) ↔ ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵)))
8	7	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵)) → (𝑦 ∈ (𝐴 ↾t 𝐵) ↔ ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵)))
9	3, 8	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵)) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵))
10	9	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ 𝑦) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵))
11		simpl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐵)) → 𝑥 ∈ 𝑦)
12		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐵)) → 𝑦 = (𝑧 ∩ 𝐵))
13	11, 12	eleqtrd 2865	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐵)) → 𝑥 ∈ (𝑧 ∩ 𝐵))
14	13	ex 417	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑦 → (𝑦 = (𝑧 ∩ 𝐵) → 𝑥 ∈ (𝑧 ∩ 𝐵)))
15	14	3ad2ant3 1153	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ 𝑦) → (𝑦 = (𝑧 ∩ 𝐵) → 𝑥 ∈ (𝑧 ∩ 𝐵)))
16	15	reximdv 3180	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ 𝑦) → (∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵) → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵)))
17	10, 16	mpd 16	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ 𝑦) → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵))
18	17	3exp 1137	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐴 ↾t 𝐵) → (𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵))))
19	18	rexlimdv 3164	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵)))
20	19	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) → (∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵)))
21	2, 20	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵))
22		elinel1 4155	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ 𝑧)
23	22	adantl 486	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ 𝑧)
24		simpl 487	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑧 ∈ 𝐴)
25		elunii 4878	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → 𝑥 ∈ ∪ 𝐴)
26	23, 24, 25	syl2anc 595	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ ∪ 𝐴)
27		elinel2 4156	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ 𝐵)
28	27	adantl 486	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ 𝐵)
29	26, 28	elind 4154	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
30	29	ex 417	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → (𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)))
31	30	adantl 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) ∧ 𝑧 ∈ 𝐴) → (𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)))
32	31	rexlimdva 3166	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) → (∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)))
33	21, 32	mpd 16	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
34	33	ralrimiva 3157	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ∪ (𝐴 ↾t 𝐵)𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
35		dfss3 3927	. . 3 ⊢ (∪ (𝐴 ↾t 𝐵) ⊆ (∪ 𝐴 ∩ 𝐵) ↔ ∀𝑥 ∈ ∪ (𝐴 ↾t 𝐵)𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
36	34, 35	sylibr 237	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) ⊆ (∪ 𝐴 ∩ 𝐵))
37		elinel1 4155	. . . . . 6 ⊢ (𝑥 ∈ (∪ 𝐴 ∩ 𝐵) → 𝑥 ∈ ∪ 𝐴)
38		eluni2 4877	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧)
39	37, 38	sylib 221	. . . . 5 ⊢ (𝑥 ∈ (∪ 𝐴 ∩ 𝐵) → ∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧)
40	39	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → ∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧)
41	4	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐴 ∈ 𝑉)
42	5	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐵 ∈ 𝑊)
43		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
44		eqid 2763	. . . . . . . . . 10 ⊢ (𝑧 ∩ 𝐵) = (𝑧 ∩ 𝐵)
45	41, 42, 43, 44	elrestd 45806	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵))
46	45	3adant3 1150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → (𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵))
47	46	3adant1r 1196	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → (𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵))
48		simp3 1156	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝑧)
49		simp1r 1217	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
50		elinel2 4156	. . . . . . . . 9 ⊢ (𝑥 ∈ (∪ 𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐵)
51	49, 50	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝐵)
52		simpl 487	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝑧)
53		simpr 489	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
54	52, 53	elind 4154	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝑧 ∩ 𝐵))
55	48, 51, 54	syl2anc 595	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ (𝑧 ∩ 𝐵))
56		eleq2 2852	. . . . . . . 8 ⊢ (𝑦 = (𝑧 ∩ 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑧 ∩ 𝐵)))
57	56	rspcev 3582	. . . . . . 7 ⊢ (((𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
58	47, 55, 57	syl2anc 595	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
59	58	3exp 1137	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → (𝑧 ∈ 𝐴 → (𝑥 ∈ 𝑧 → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)))
60	59	rexlimdv 3164	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → (∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧 → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦))
61	40, 60	mpd 16	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
62	61, 1	sylibr 237	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → 𝑥 ∈ ∪ (𝐴 ↾t 𝐵))
63	36, 62	eqelssd 3959	1 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ∩ cin 3905   ⊆ wss 3906  ∪ cuni 4873  (class class class)co 7412   ↾_t crest 17474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-rest 17476

This theorem is referenced by:  restuni4  45819  subsalsal  47053