Theorem ussid 24204

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 7366	. . 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 6848	. . . . . . 7 ⊢ 𝐵 ∈ V
5	4, 4	xpex 7698	. . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V
6	2, 5	eqeltrrdi 2845	. . . . 5 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → ∪ 𝑈 ∈ V)
7		uniexb 7709	. . . . 5 ⊢ (𝑈 ∈ V ↔ ∪ 𝑈 ∈ V)
8	6, 7	sylibr 234	. . . 4 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 ∈ V)
9		eqid 2736	. . . . 5 ⊢ ∪ 𝑈 = ∪ 𝑈
10	9	restid 17353	. . . 4 ⊢ (𝑈 ∈ V → (𝑈 ↾t ∪ 𝑈) = 𝑈)
11	8, 10	syl 17	. . 3 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → (𝑈 ↾t ∪ 𝑈) = 𝑈)
12	1, 11	eqtr2d 2772	. 2 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (𝑈 ↾t (𝐵 × 𝐵)))
13		ussval.2	. . 3 ⊢ 𝑈 = (UnifSet‘𝑊)
14	3, 13	ussval 24203	. 2 ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊)
15	12, 14	eqtrdi 2787	1 ⊢ ((𝐵 × 𝐵) = ∪ 𝑈 → 𝑈 = (UnifSt‘𝑊))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ∪ cuni 4863   × cxp 5622  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  UnifSetcunif 17187   ↾_t crest 17340  UnifStcuss 24197

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-rest 17342  df-uss 24200

This theorem is referenced by:  tususs  24213  cnflduss  25312