Theorem ussid 24269

Description: In case the base of the UnifSt element of the uniform space is the base of its element structure, then UnifSt does not restrict it further. (Contributed by Thierry Arnoux, 4-Dec-2017.)

Hypotheses

Ref	Expression
ussval.1	⊢ 𝐵 = (Base‘𝑊)
ussval.2	⊢ 𝑈 = (UnifSet‘𝑊)

Assertion

Ref	Expression
ussid	⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (UnifSt‘𝑊))

Proof of Theorem ussid

Step	Hyp	Ref	Expression
1		oveq2 7439	. . 3 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → (𝑈 ↾t (𝐵 × 𝐵)) = (𝑈 ↾t ∪ 𝑈))
2		id 22	. . . . . 6 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → (𝐵 × 𝐵) = ∪ 𝑈)
3		ussval.1	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑊)
4	3	fvexi 6920	. . . . . . 7 ⊢ 𝐵 ∈ V
5	4, 4	xpex 7773	. . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V
6	2, 5	eqeltrrdi 2850	. . . . 5 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → ∪ 𝑈 ∈ V)
7		uniexb 7784	. . . . 5 ⊢ (𝑈 ∈ V ↔ ∪ 𝑈 ∈ V)
8	6, 7	sylibr 234	. . . 4 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 ∈ V)
9		eqid 2737	. . . . 5 ⊢ ∪ 𝑈 = ∪ 𝑈
10	9	restid 17478	. . . 4 ⊢ (𝑈 ∈ V → (𝑈 ↾t ∪ 𝑈) = 𝑈)
11	8, 10	syl 17	. . 3 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → (𝑈 ↾t ∪ 𝑈) = 𝑈)
12	1, 11	eqtr2d 2778	. 2 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (𝑈 ↾t (𝐵 × 𝐵)))
13		ussval.2	. . 3 ⊢ 𝑈 = (UnifSet‘𝑊)
14	3, 13	ussval 24268	. 2 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊)
15	12, 14	eqtrdi 2793	1 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (UnifSt‘𝑊))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ∪ cuni 4907   × cxp 5683  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  UnifSetcunif 17307   ↾_t crest 17465  UnifStcuss 24262

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-pow 5365  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-pw 4602  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-1st 8014  df-2nd 8015  df-rest 17467  df-uss 24265

This theorem is referenced by:  tususs  24279  cnflduss  25390