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Theorem ucnprima 23179
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 23178 . . 3 (𝜑 → ∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊)
75mpofun 7334 . . . . 5 Fun 𝐺
8 ustssxp 23102 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
91, 8sylan 583 . . . . . 6 ((𝜑𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
10 opex 5348 . . . . . . 7 ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ V
115, 10dmmpo 7841 . . . . . 6 dom 𝐺 = (𝑋 × 𝑋)
129, 11sseqtrrdi 3952 . . . . 5 ((𝜑𝑟𝑈) → 𝑟 ⊆ dom 𝐺)
13 funimass3 6874 . . . . 5 ((Fun 𝐺𝑟 ⊆ dom 𝐺) → ((𝐺𝑟) ⊆ 𝑊𝑟 ⊆ (𝐺𝑊)))
147, 12, 13sylancr 590 . . . 4 ((𝜑𝑟𝑈) → ((𝐺𝑟) ⊆ 𝑊𝑟 ⊆ (𝐺𝑊)))
1514rexbidva 3215 . . 3 (𝜑 → (∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊 ↔ ∃𝑟𝑈 𝑟 ⊆ (𝐺𝑊)))
166, 15mpbid 235 . 2 (𝜑 → ∃𝑟𝑈 𝑟 ⊆ (𝐺𝑊))
171adantr 484 . . . 4 ((𝜑𝑟𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
18 simpr 488 . . . 4 ((𝜑𝑟𝑈) → 𝑟𝑈)
19 cnvimass 5949 . . . . . 6 (𝐺𝑊) ⊆ dom 𝐺
2019, 11sseqtri 3937 . . . . 5 (𝐺𝑊) ⊆ (𝑋 × 𝑋)
2120a1i 11 . . . 4 ((𝜑𝑟𝑈) → (𝐺𝑊) ⊆ (𝑋 × 𝑋))
22 ustssel 23103 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟𝑈 ∧ (𝐺𝑊) ⊆ (𝑋 × 𝑋)) → (𝑟 ⊆ (𝐺𝑊) → (𝐺𝑊) ∈ 𝑈))
2317, 18, 21, 22syl3anc 1373 . . 3 ((𝜑𝑟𝑈) → (𝑟 ⊆ (𝐺𝑊) → (𝐺𝑊) ∈ 𝑈))
2423rexlimdva 3203 . 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 209  wa 399   = wceq 1543  wcel 2110  wrex 3062  wss 3866  cop 4547   × cxp 5549  ccnv 5550  dom cdm 5551  cima 5554  Fun wfun 6374  cfv 6380  (class class class)co 7213  cmpo 7215  UnifOncust 23097   Cnucucn 23172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-map 8510  df-ust 23098  df-ucn 23173
This theorem is referenced by:  fmucnd  23189
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