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Theorem elutop 22842
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 22841 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
21eleq2d 2878 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ 𝐴 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎}))
3 sseq2 3944 . . . . . 6 (𝑎 = 𝐴 → ((𝑣 “ {𝑥}) ⊆ 𝑎 ↔ (𝑣 “ {𝑥}) ⊆ 𝐴))
43rexbidv 3259 . . . . 5 (𝑎 = 𝐴 → (∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))
54raleqbi1dv 3359 . . . 4 (𝑎 = 𝐴 → (∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∀𝑥𝐴𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))
65elrab 3631 . . 3 (𝐴 ∈ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ↔ (𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑥𝐴𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴))
72, 6syl6bb 290 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑥𝐴𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))
8 elex 3462 . . . . 5 (𝐴 ∈ 𝒫 𝑋𝐴 ∈ V)
98a1i 11 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋𝐴 ∈ V))
10 elfvex 6682 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
1110adantr 484 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝑋 ∈ V)
12 simpr 488 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
1311, 12ssexd 5195 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
1413ex 416 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴𝑋𝐴 ∈ V))
15 elpwg 4503 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1615a1i 11 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋𝐴𝑋)))
179, 14, 16pm5.21ndd 384 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1817anbi1d 632 . 2 (𝑈 ∈ (UnifOn‘𝑋) → ((𝐴 ∈ 𝒫 𝑋 ∧ ∀𝑥𝐴𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴) ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))
197, 18bitrd 282 1 (𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  wss 3884  𝒫 cpw 4500  {csn 4528  cima 5526  cfv 6328  UnifOncust 22808  unifTopcutop 22839
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-ust 22809  df-utop 22840
This theorem is referenced by:  utoptop  22843  utopbas  22844  restutop  22846  restutopopn  22847  ucncn  22894
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