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Theorem elutop 22256
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 22255 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
21eleq2d 2835 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ 𝐴 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎}))
3 sseq2 3774 . . . . . 6 (𝑎 = 𝐴 → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ (𝑣 “ {𝑥}) ⊆ 𝐴))
43rexbidv 3199 . . . . 5 (𝑎 = 𝐴 → (∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))
54raleqbi1dv 3294 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∀𝑥𝐴𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))
65elrab 3513 . . 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 3361 . . . . 5 (𝐴 ∈ 𝒫 𝑋𝐴 ∈ V)
98a1i 11 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋𝐴 ∈ V))
10 elfvex 6362 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
1110adantr 466 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝑋 ∈ V)
12 simpr 471 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
1311, 12ssexd 4936 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
1413ex 397 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴𝑋𝐴 ∈ V))
15 elpwg 4303 . . . . 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 607 . 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 382   = wceq 1630  wcel 2144  wral 3060  wrex 3061  {crab 3064  Vcvv 3349  wss 3721  𝒫 cpw 4295  {csn 4314  cima 5252  cfv 6031  UnifOncust 22222  unifTopcutop 22253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-ust 22223  df-utop 22254
This theorem is referenced by:  utoptop  22257  utopbas  22258  restutop  22260  restutopopn  22261  ucncn  22308
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