Theorem utopbas 24191

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 24188	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
2		ssrab2 4034	. . . 4 ⊢ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ⊆ 𝒫 𝑋
3	1, 2	eqsstrdi 3980	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
4		ssidd 3959	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ⊆ 𝑋)
5		ustssxp 24161	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑣 ⊆ (𝑋 × 𝑋))
6		imassrn 6038	. . . . . . . . . 10 ⊢ (𝑣 “ {𝑥}) ⊆ ran 𝑣
7		rnss 5896	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣 ⊆ ran (𝑋 × 𝑋))
8		rnxpid 6139	. . . . . . . . . . 11 ⊢ ran (𝑋 × 𝑋) = 𝑋
9	7, 8	sseqtrdi 3976	. . . . . . . . . 10 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣 ⊆ 𝑋)
10	6, 9	sstrid 3947	. . . . . . . . 9 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → (𝑣 “ {𝑥}) ⊆ 𝑋)
11	5, 10	syl 17	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑥}) ⊆ 𝑋)
12	11	ralrimiva 3130	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
13		ustne0 24170	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
14		r19.2zb 4455	. . . . . . . 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 3134	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥 ∈ 𝑋 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
18		elutop 24189	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 ∈ (unifTop‘𝑈) ↔ (𝑋 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑋 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)))
19	4, 17, 18	mpbir2and 714	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ (unifTop‘𝑈))
20		elpwuni 5062	. . . 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 2743	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3401   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  {csn 4582  ∪ cuni 4865   × cxp 5630  ran crn 5633   “ cima 5635  ‘cfv 6500  UnifOncust 24156  unifTopcutop 24186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fv 6508  df-ust 24157  df-utop 24187

This theorem is referenced by:  utoptopon  24192  utop2nei  24206  utopreg  24208  tuslem  24222