Theorem ussid 24216

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 7376	. . 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 6856	. . . . . . 7 ⊢ 𝐵 ∈ V
5	4, 4	xpex 7708	. . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V
6	2, 5	eqeltrrdi 2846	. . . . 5 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → ∪ 𝑈 ∈ V)
7		uniexb 7719	. . . . 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 17365	. . . 4 ⊢ (𝑈 ∈ V → (𝑈 ↾t ∪ 𝑈) = 𝑈)
11	8, 10	syl 17	. . 3 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → (𝑈 ↾t ∪ 𝑈) = 𝑈)
12	1, 11	eqtr2d 2773	. 2 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (𝑈 ↾t (𝐵 × 𝐵)))
13		ussval.2	. . 3 ⊢ 𝑈 = (UnifSet‘𝑊)
14	3, 13	ussval 24215	. 2 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊)
15	12, 14	eqtrdi 2788	1 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (UnifSt‘𝑊))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∪ cuni 4865   × cxp 5630  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  UnifSetcunif 17199   ↾_t crest 17352  UnifStcuss 24209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-rest 17354  df-uss 24212

This theorem is referenced by:  tususs  24225  cnflduss  25324