Theorem utopbas 23387

Description: The base of the topology induced by a uniform structure 𝑈. (Contributed by Thierry Arnoux, 5-Dec-2017.)

Assertion

Ref	Expression
utopbas	⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))

Proof of Theorem utopbas

Dummy variables 𝑎 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		utopval 23384	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
2		ssrab2 4013	. . . 4 ⊢ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ⊆ 𝒫 𝑋
3	1, 2	eqsstrdi 3975	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
4		ssidd 3944	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ⊆ 𝑋)
5		ustssxp 23356	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑣 ⊆ (𝑋 × 𝑋))
6		imassrn 5980	. . . . . . . . . 10 ⊢ (𝑣 “ {𝑥}) ⊆ ran 𝑣
7		rnss 5848	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣 ⊆ ran (𝑋 × 𝑋))
8		rnxpid 6076	. . . . . . . . . . 11 ⊢ ran (𝑋 × 𝑋) = 𝑋
9	7, 8	sseqtrdi 3971	. . . . . . . . . 10 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣 ⊆ 𝑋)
10	6, 9	sstrid 3932	. . . . . . . . 9 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → (𝑣 “ {𝑥}) ⊆ 𝑋)
11	5, 10	syl 17	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑥}) ⊆ 𝑋)
12	11	ralrimiva 3103	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
13		ustne0 23365	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
14		r19.2zb 4426	. . . . . . . 8 ⊢ (𝑈 ≠ ∅ ↔ (∀𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋 → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋))
15	13, 14	sylib 217	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (∀𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋 → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋))
16	12, 15	mpd 15	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
17	16	ralrimivw 3104	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥 ∈ 𝑋 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
18		elutop 23385	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 ∈ (unifTop‘𝑈) ↔ (𝑋 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)))
19	4, 17, 18	mpbir2and 710	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ (unifTop‘𝑈))
20		elpwuni 5034	. . . 4 ⊢ (𝑋 ∈ (unifTop‘𝑈) → ((unifTop‘𝑈) ⊆ 𝒫 𝑋 ↔ ∪ (unifTop‘𝑈) = 𝑋))
21	19, 20	syl 17	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((unifTop‘𝑈) ⊆ 𝒫 𝑋 ↔ ∪ (unifTop‘𝑈) = 𝑋))
22	3, 21	mpbid 231	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (unifTop‘𝑈) = 𝑋)
23	22	eqcomd 2744	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561  ∪ cuni 4839   × cxp 5587  ran crn 5590   “ cima 5592  ‘cfv 6433  UnifOncust 23351  unifTopcutop 23382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-ust 23352  df-utop 23383

This theorem is referenced by:  utoptopon  23388  utop2nei  23402  utopreg  23404  tuslem  23418  tuslemOLD  23419