Theorem utopbas 24179

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 24176	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
2		ssrab2 4032	. . . 4 ⊢ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ⊆ 𝒫 𝑋
3	1, 2	eqsstrdi 3978	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
4		ssidd 3957	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ⊆ 𝑋)
5		ustssxp 24149	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑣 ⊆ (𝑋 × 𝑋))
6		imassrn 6030	. . . . . . . . . 10 ⊢ (𝑣 “ {𝑥}) ⊆ ran 𝑣
7		rnss 5888	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣 ⊆ ran (𝑋 × 𝑋))
8		rnxpid 6131	. . . . . . . . . . 11 ⊢ ran (𝑋 × 𝑋) = 𝑋
9	7, 8	sseqtrdi 3974	. . . . . . . . . 10 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → ran 𝑣 ⊆ 𝑋)
10	6, 9	sstrid 3945	. . . . . . . . 9 ⊢ (𝑣 ⊆ (𝑋 × 𝑋) → (𝑣 “ {𝑥}) ⊆ 𝑋)
11	5, 10	syl 17	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → (𝑣 “ {𝑥}) ⊆ 𝑋)
12	11	ralrimiva 3128	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
13		ustne0 24158	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
14		r19.2zb 4453	. . . . . . . 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 3132	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥 ∈ 𝑋 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑋)
18		elutop 24177	. . . . 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 5060	. . . 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 2742	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3399   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  {csn 4580  ∪ cuni 4863   × cxp 5622  ran crn 5625   “ cima 5627  ‘cfv 6492  UnifOncust 24144  unifTopcutop 24174

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-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-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-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-fv 6500  df-ust 24145  df-utop 24175

This theorem is referenced by:  utoptopon  24180  utop2nei  24194  utopreg  24196  tuslem  24210