Theorem restuni3 45660

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 4868	. . . . . . 7 ⊢ (𝑥 ∈ ∪ (𝐴 ↾t 𝐵) ↔ ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
2	1	bilani 508	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
3		simpr 488	. . . . . . . . . . . 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 17439	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑦 ∈ (𝐴 ↾t 𝐵) ↔ ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵)))
7	4, 5, 6	syl2anc 593	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑦 ∈ (𝐴 ↾t 𝐵) ↔ ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵)))
8	7	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵)) → (𝑦 ∈ (𝐴 ↾t 𝐵) ↔ ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵)))
9	3, 8	mpbid 234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵)) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵))
10	9	3adant3 1144	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ 𝑦) → ∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵))
11		simpl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐵)) → 𝑥 ∈ 𝑦)
12		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐵)) → 𝑦 = (𝑧 ∩ 𝐵))
13	11, 12	eleqtrd 2863	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (𝑧 ∩ 𝐵)) → 𝑥 ∈ (𝑧 ∩ 𝐵))
14	13	ex 416	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑦 → (𝑦 = (𝑧 ∩ 𝐵) → 𝑥 ∈ (𝑧 ∩ 𝐵)))
15	14	3ad2ant3 1147	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ 𝑦) → (𝑦 = (𝑧 ∩ 𝐵) → 𝑥 ∈ (𝑧 ∩ 𝐵)))
16	15	reximdv 3176	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ 𝑦) → (∃𝑧 ∈ 𝐴 𝑦 = (𝑧 ∩ 𝐵) → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵)))
17	10, 16	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ 𝑦) → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵))
18	17	3exp 1131	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐴 ↾t 𝐵) → (𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵))))
19	18	rexlimdv 3160	. . . . . . 7 ⊢ (𝜑 → (∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵)))
20	19	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) → (∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵)))
21	2, 20	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) → ∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵))
22		elinel1 4153	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ 𝑧)
23	22	adantl 485	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ 𝑧)
24		simpl 486	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑧 ∈ 𝐴)
25		elunii 4869	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴) → 𝑥 ∈ ∪ 𝐴)
26	23, 24, 25	syl2anc 593	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ ∪ 𝐴)
27		elinel2 4154	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ 𝐵)
28	27	adantl 485	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ 𝐵)
29	26, 28	elind 4152	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
30	29	ex 416	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → (𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)))
31	30	adantl 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) ∧ 𝑧 ∈ 𝐴) → (𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)))
32	31	rexlimdva 3162	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) → (∃𝑧 ∈ 𝐴 𝑥 ∈ (𝑧 ∩ 𝐵) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)))
33	21, 32	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝐴 ↾t 𝐵)) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
34	33	ralrimiva 3153	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ∪ (𝐴 ↾t 𝐵)𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
35		dfss3 3925	. . 3 ⊢ (∪ (𝐴 ↾t 𝐵) ⊆ (∪ 𝐴 ∩ 𝐵) ↔ ∀𝑥 ∈ ∪ (𝐴 ↾t 𝐵)𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
36	34, 35	sylibr 236	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) ⊆ (∪ 𝐴 ∩ 𝐵))
37		elinel1 4153	. . . . . 6 ⊢ (𝑥 ∈ (∪ 𝐴 ∩ 𝐵) → 𝑥 ∈ ∪ 𝐴)
38		eluni2 4868	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧)
39	37, 38	sylib 220	. . . . 5 ⊢ (𝑥 ∈ (∪ 𝐴 ∩ 𝐵) → ∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧)
40	39	adantl 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → ∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧)
41	4	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐴 ∈ 𝑉)
42	5	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐵 ∈ 𝑊)
43		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
44		eqid 2761	. . . . . . . . . 10 ⊢ (𝑧 ∩ 𝐵) = (𝑧 ∩ 𝐵)
45	41, 42, 43, 44	elrestd 45650	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵))
46	45	3adant3 1144	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → (𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵))
47	46	3adant1r 1190	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → (𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵))
48		simp3 1150	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝑧)
49		simp1r 1211	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ (∪ 𝐴 ∩ 𝐵))
50		elinel2 4154	. . . . . . . . 9 ⊢ (𝑥 ∈ (∪ 𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐵)
51	49, 50	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ 𝐵)
52		simpl 486	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝑧)
53		simpr 488	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
54	52, 53	elind 4152	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝑧 ∩ 𝐵))
55	48, 51, 54	syl2anc 593	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → 𝑥 ∈ (𝑧 ∩ 𝐵))
56		eleq2 2850	. . . . . . . 8 ⊢ (𝑦 = (𝑧 ∩ 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑧 ∩ 𝐵)))
57	56	rspcev 3581	. . . . . . 7 ⊢ (((𝑧 ∩ 𝐵) ∈ (𝐴 ↾t 𝐵) ∧ 𝑥 ∈ (𝑧 ∩ 𝐵)) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
58	47, 55, 57	syl2anc 593	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑥 ∈ 𝑧) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
59	58	3exp 1131	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → (𝑧 ∈ 𝐴 → (𝑥 ∈ 𝑧 → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)))
60	59	rexlimdv 3160	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → (∃𝑧 ∈ 𝐴 𝑥 ∈ 𝑧 → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦))
61	40, 60	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → ∃𝑦 ∈ (𝐴 ↾t 𝐵)𝑥 ∈ 𝑦)
62	61, 1	sylibr 236	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (∪ 𝐴 ∩ 𝐵)) → 𝑥 ∈ ∪ (𝐴 ↾t 𝐵))
63	36, 62	eqelssd 3957	1 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085   ∩ cin 3903   ⊆ wss 3904  ∪ cuni 4864  (class class class)co 7392   ↾_t crest 17432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-rest 17434

This theorem is referenced by:  restuni4  45663  subsalsal  46897