Theorem utopbas 24153

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 24150	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
2		ssrab2 4029	. . . 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 3954	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ⊆ 𝑋)
5		ustssxp 24123	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑣 ⊆ (𝑋 × 𝑋))
6		imassrn 6026	. . . . . . . . . 10 ⊢ (𝑣 “ {𝑥}) ⊆ ran 𝑣
7		rnss 5885	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣 ⊆ ran (𝑋 × 𝑋))
8		rnxpid 6127	. . . . . . . . . . 11 ⊢ ran (𝑋 × 𝑋) = 𝑋
9	7, 8	sseqtrdi 3971	. . . . . . . . . 10 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣 ⊆ 𝑋)
10	6, 9	sstrid 3942	. . . . . . . . 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 24132	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
14		r19.2zb 4450	. . . . . . . 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 24151	. . . . 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 5057	. . . 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 2739	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  ∃wrex 3057  {crab 3396   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4551  {csn 4577  ∪ cuni 4860   × cxp 5619  ran crn 5622   “ cima 5624  ‘cfv 6488  UnifOncust 24118  unifTopcutop 24148

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-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

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-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-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 6444  df-fun 6490  df-fv 6496  df-ust 24119  df-utop 24149

This theorem is referenced by:  utoptopon  24154  utop2nei  24168  utopreg  24170  tuslem  24184