Theorem elutop 24267

Description: Open sets in the topology induced by an uniform structure 𝑈 on 𝑋 (Contributed by Thierry Arnoux, 30-Nov-2017.)

Assertion

Ref	Expression
elutop	⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))

Distinct variable groups:   𝑥,𝑣,𝐴   𝑣,𝑈,𝑥   𝑥,𝑋

Allowed substitution hint:   𝑋(𝑣)

Proof of Theorem elutop

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		utopval 24266	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
2	1	eleq2d 2827	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ 𝐴 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎}))
3		sseq2 4025	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ (𝑣 “ {𝑥}) ⊆ 𝐴))
4	3	rexbidv 3179	. . . . 5 ⊢ (𝑎 = 𝐴 → (∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))
5	4	raleqbi1dv 3338	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))
6	5	elrab 3698	. . 3 ⊢ (𝐴 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ↔ (𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))
7	2, 6	bitrdi 287	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))
8		elex 3502	. . . . 5 ⊢ (𝐴 ∈ 𝒫 𝑋 → 𝐴 ∈ V)
9	8	a1i 11	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋 → 𝐴 ∈ V))
10		elfvex 6952	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
11	10	adantr 480	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 ∈ V)
12		simpr 484	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋)
13	11, 12	ssexd 5333	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ V)
14	13	ex 412	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ⊆ 𝑋 → 𝐴 ∈ V))
15		elpwg 4611	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
16	15	a1i 11	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋)))
17	9, 14, 16	pm5.21ndd 379	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
18	17	anbi1d 631	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))
19	7, 18	bitrd 279	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3481   ⊆ wss 3966  𝒫 cpw 4608  {csn 4634   “ cima 5696  ‘cfv 6569  UnifOncust 24233  unifTopcutop 24264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-iota 6522  df-fun 6571  df-fv 6577  df-ust 24234  df-utop 24265

This theorem is referenced by:  utoptop  24268  utopbas  24269  restutop  24271  restutopopn  24272  ucncn  24319