MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ucnprima Structured version   Visualization version   GIF version

Theorem ucnprima 22133
Description: The preimage by a uniformly continuous function 𝐹 of an entourage 𝑊 of 𝑌 is an entourage of 𝑋. Note of the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 19-Nov-2017.)
Hypotheses
Ref Expression
ucnprima.1 (𝜑𝑈 ∈ (UnifOn‘𝑋))
ucnprima.2 (𝜑𝑉 ∈ (UnifOn‘𝑌))
ucnprima.3 (𝜑𝐹 ∈ (𝑈 Cnu𝑉))
ucnprima.4 (𝜑𝑊𝑉)
ucnprima.5 𝐺 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
Assertion
Ref Expression
ucnprima (𝜑 → (𝐺𝑊) ∈ 𝑈)
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝑋,𝑦   𝑥,𝐺,𝑦   𝑥,𝑈,𝑦   𝑥,𝑉   𝑥,𝑊,𝑦   𝑥,𝑌   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑉(𝑦)   𝑌(𝑦)

Proof of Theorem ucnprima
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 ucnprima.1 . . . 4 (𝜑𝑈 ∈ (UnifOn‘𝑋))
2 ucnprima.2 . . . 4 (𝜑𝑉 ∈ (UnifOn‘𝑌))
3 ucnprima.3 . . . 4 (𝜑𝐹 ∈ (𝑈 Cnu𝑉))
4 ucnprima.4 . . . 4 (𝜑𝑊𝑉)
5 ucnprima.5 . . . 4 𝐺 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
61, 2, 3, 4, 5ucnima 22132 . . 3 (𝜑 → ∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊)
75mpt2fun 6804 . . . . 5 Fun 𝐺
8 ustssxp 22055 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
91, 8sylan 487 . . . . . 6 ((𝜑𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
10 opex 4962 . . . . . . 7 ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ V
115, 10dmmpt2 7285 . . . . . 6 dom 𝐺 = (𝑋 × 𝑋)
129, 11syl6sseqr 3685 . . . . 5 ((𝜑𝑟𝑈) → 𝑟 ⊆ dom 𝐺)
13 funimass3 6373 . . . . 5 ((Fun 𝐺𝑟 ⊆ dom 𝐺) → ((𝐺𝑟) ⊆ 𝑊𝑟 ⊆ (𝐺𝑊)))
147, 12, 13sylancr 696 . . . 4 ((𝜑𝑟𝑈) → ((𝐺𝑟) ⊆ 𝑊𝑟 ⊆ (𝐺𝑊)))
1514rexbidva 3078 . . 3 (𝜑 → (∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊 ↔ ∃𝑟𝑈 𝑟 ⊆ (𝐺𝑊)))
166, 15mpbid 222 . 2 (𝜑 → ∃𝑟𝑈 𝑟 ⊆ (𝐺𝑊))
171adantr 480 . . . 4 ((𝜑𝑟𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
18 simpr 476 . . . 4 ((𝜑𝑟𝑈) → 𝑟𝑈)
19 cnvimass 5520 . . . . . 6 (𝐺𝑊) ⊆ dom 𝐺
2019, 11sseqtri 3670 . . . . 5 (𝐺𝑊) ⊆ (𝑋 × 𝑋)
2120a1i 11 . . . 4 ((𝜑𝑟𝑈) → (𝐺𝑊) ⊆ (𝑋 × 𝑋))
22 ustssel 22056 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟𝑈 ∧ (𝐺𝑊) ⊆ (𝑋 × 𝑋)) → (𝑟 ⊆ (𝐺𝑊) → (𝐺𝑊) ∈ 𝑈))
2317, 18, 21, 22syl3anc 1366 . . 3 ((𝜑𝑟𝑈) → (𝑟 ⊆ (𝐺𝑊) → (𝐺𝑊) ∈ 𝑈))
2423rexlimdva 3060 . 2 (𝜑 → (∃𝑟𝑈 𝑟 ⊆ (𝐺𝑊) → (𝐺𝑊) ∈ 𝑈))
2516, 24mpd 15 1 (𝜑 → (𝐺𝑊) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  wss 3607  cop 4216   × cxp 5141  ccnv 5142  dom cdm 5143  cima 5146  Fun wfun 5920  cfv 5926  (class class class)co 6690  cmpt2 6692  UnifOncust 22050   Cnucucn 22126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-ust 22051  df-ucn 22127
This theorem is referenced by:  fmucnd  22143
  Copyright terms: Public domain W3C validator