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

Theorem ucnprima 22806
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 22805 . . 3 (𝜑 → ∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊)
75mpofun 7269 . . . . 5 Fun 𝐺
8 ustssxp 22728 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
91, 8sylan 580 . . . . . 6 ((𝜑𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
10 opex 5352 . . . . . . 7 ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ V
115, 10dmmpo 7763 . . . . . 6 dom 𝐺 = (𝑋 × 𝑋)
129, 11sseqtrrdi 4021 . . . . 5 ((𝜑𝑟𝑈) → 𝑟 ⊆ dom 𝐺)
13 funimass3 6819 . . . . 5 ((Fun 𝐺𝑟 ⊆ dom 𝐺) → ((𝐺𝑟) ⊆ 𝑊𝑟 ⊆ (𝐺𝑊)))
147, 12, 13sylancr 587 . . . 4 ((𝜑𝑟𝑈) → ((𝐺𝑟) ⊆ 𝑊𝑟 ⊆ (𝐺𝑊)))
1514rexbidva 3300 . . 3 (𝜑 → (∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊 ↔ ∃𝑟𝑈 𝑟 ⊆ (𝐺𝑊)))
166, 15mpbid 233 . 2 (𝜑 → ∃𝑟𝑈 𝑟 ⊆ (𝐺𝑊))
171adantr 481 . . . 4 ((𝜑𝑟𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
18 simpr 485 . . . 4 ((𝜑𝑟𝑈) → 𝑟𝑈)
19 cnvimass 5946 . . . . . 6 (𝐺𝑊) ⊆ dom 𝐺
2019, 11sseqtri 4006 . . . . 5 (𝐺𝑊) ⊆ (𝑋 × 𝑋)
2120a1i 11 . . . 4 ((𝜑𝑟𝑈) → (𝐺𝑊) ⊆ (𝑋 × 𝑋))
22 ustssel 22729 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟𝑈 ∧ (𝐺𝑊) ⊆ (𝑋 × 𝑋)) → (𝑟 ⊆ (𝐺𝑊) → (𝐺𝑊) ∈ 𝑈))
2317, 18, 21, 22syl3anc 1365 . . 3 ((𝜑𝑟𝑈) → (𝑟 ⊆ (𝐺𝑊) → (𝐺𝑊) ∈ 𝑈))
2423rexlimdva 3288 . 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 207  wa 396   = wceq 1530  wcel 2107  wrex 3143  wss 3939  cop 4569   × cxp 5551  ccnv 5552  dom cdm 5553  cima 5556  Fun wfun 6345  cfv 6351  (class class class)co 7151  cmpo 7153  UnifOncust 22723   Cnucucn 22799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-map 8401  df-ust 22724  df-ucn 22800
This theorem is referenced by:  fmucnd  22816
  Copyright terms: Public domain W3C validator