Theorem utopbas 24123

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 24120	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
2		ssrab2 4043	. . . 4 ⊢ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ⊆ 𝒫 𝑋
3	1, 2	eqsstrdi 3991	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
4		ssidd 3970	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ⊆ 𝑋)
5		ustssxp 24092	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑣 ⊆ (𝑋 × 𝑋))
6		imassrn 6042	. . . . . . . . . 10 ⊢ (𝑣 “ {𝑥}) ⊆ ran 𝑣
7		rnss 5903	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣 ⊆ ran (𝑋 × 𝑋))
8		rnxpid 6146	. . . . . . . . . . 11 ⊢ ran (𝑋 × 𝑋) = 𝑋
9	7, 8	sseqtrdi 3987	. . . . . . . . . 10 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣 ⊆ 𝑋)
10	6, 9	sstrid 3958	. . . . . . . . 9 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → (𝑣 “ {𝑥}) ⊆ 𝑋)
11	5, 10	syl 17	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑥}) ⊆ 𝑋)
12	11	ralrimiva 3125	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
13		ustne0 24101	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
14		r19.2zb 4459	. . . . . . . 8 ⊢ (𝑈 ≠ ∅ ↔ (∀𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋 → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋))
15	13, 14	sylib 218	. . . . . . 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 3129	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥 ∈ 𝑋 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
18		elutop 24121	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 ∈ (unifTop‘𝑈) ↔ (𝑋 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)))
19	4, 17, 18	mpbir2and 713	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ (unifTop‘𝑈))
20		elpwuni 5069	. . . 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 232	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (unifTop‘𝑈) = 𝑋)
23	22	eqcomd 2735	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3405   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  {csn 4589  ∪ cuni 4871   × cxp 5636  ran crn 5639   “ cima 5641  ‘cfv 6511  UnifOncust 24087  unifTopcutop 24118

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-ust 24088  df-utop 24119

This theorem is referenced by:  utoptopon  24124  utop2nei  24138  utopreg  24140  tuslem  24154