Theorem ussid 24195

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 7363	. . 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 6845	. . . . . . 7 ⊢ 𝐵 ∈ V
5	4, 4	xpex 7695	. . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V
6	2, 5	eqeltrrdi 2842	. . . . 5 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → ∪ 𝑈 ∈ V)
7		uniexb 7706	. . . . 5 ⊢ (𝑈 ∈ V ↔ ∪ 𝑈 ∈ V)
8	6, 7	sylibr 234	. . . 4 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 ∈ V)
9		eqid 2733	. . . . 5 ⊢ ∪ 𝑈 = ∪ 𝑈
10	9	restid 17344	. . . 4 ⊢ (𝑈 ∈ V → (𝑈 ↾t ∪ 𝑈) = 𝑈)
11	8, 10	syl 17	. . 3 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → (𝑈 ↾t ∪ 𝑈) = 𝑈)
12	1, 11	eqtr2d 2769	. 2 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (𝑈 ↾t (𝐵 × 𝐵)))
13		ussval.2	. . 3 ⊢ 𝑈 = (UnifSet‘𝑊)
14	3, 13	ussval 24194	. 2 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊)
15	12, 14	eqtrdi 2784	1 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (UnifSt‘𝑊))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ∪ cuni 4860   × cxp 5619  ‘cfv 6489  (class class class)co 7355  Basecbs 17127  UnifSetcunif 17178   ↾_t crest 17331  UnifStcuss 24188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-rest 17333  df-uss 24191

This theorem is referenced by:  tususs  24204  cnflduss  25303