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Theorem elutop 22127
 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) ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑣𝑈 (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.
StepHypRef Expression
1 utopval 22126 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
21eleq2d 2757 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ 𝐴 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎}))
3 sseq2 3701 . . . . . 6 (𝑎 = 𝐴 → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ (𝑣 “ {𝑥}) ⊆ 𝐴))
43rexbidv 3122 . . . . 5 (𝑎 = 𝐴 → (∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))
54raleqbi1dv 3217 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∀𝑥𝐴𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))
65elrab 3437 . . 3 (𝐴 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ↔ (𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑥𝐴𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))
72, 6syl6bb 276 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑥𝐴𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))
8 elex 3284 . . . . 5 (𝐴 ∈ 𝒫 𝑋𝐴 ∈ V)
98a1i 11 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋𝐴 ∈ V))
10 elfvex 6302 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
1110adantr 472 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝑋 ∈ V)
12 simpr 479 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
1311, 12ssexd 4881 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
1413ex 449 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴𝑋𝐴 ∈ V))
15 elpwg 4242 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1615a1i 11 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋𝐴𝑋)))
179, 14, 16pm5.21ndd 368 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1817anbi1d 743 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ((𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑥𝐴𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴) ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))
197, 18bitrd 268 1 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1564   ∈ wcel 2071  ∀wral 2982  ∃wrex 2983  {crab 2986  Vcvv 3272   ⊆ wss 3648  𝒫 cpw 4234  {csn 4253   “ cima 5189  ‘cfv 5969  UnifOncust 22093  unifTopcutop 22124 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-rab 2991  df-v 3274  df-sbc 3510  df-csb 3608  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-op 4260  df-uni 4513  df-br 4729  df-opab 4789  df-mpt 4806  df-id 5096  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-iota 5932  df-fun 5971  df-fn 5972  df-fv 5977  df-ust 22094  df-utop 22125 This theorem is referenced by:  utoptop  22128  utopbas  22129  restutop  22131  restutopopn  22132  ucncn  22179
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