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Theorem ucnprima 22898
 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 22897 . . 3 (𝜑 → ∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊)
75mpofun 7256 . . . . 5 Fun 𝐺
8 ustssxp 22820 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
91, 8sylan 583 . . . . . 6 ((𝜑𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
10 opex 5322 . . . . . . 7 ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ V
115, 10dmmpo 7754 . . . . . 6 dom 𝐺 = (𝑋 × 𝑋)
129, 11sseqtrrdi 3966 . . . . 5 ((𝜑𝑟𝑈) → 𝑟 ⊆ dom 𝐺)
13 funimass3 6802 . . . . 5 ((Fun 𝐺𝑟 ⊆ dom 𝐺) → ((𝐺𝑟) ⊆ 𝑊𝑟 ⊆ (𝐺𝑊)))
147, 12, 13sylancr 590 . . . 4 ((𝜑𝑟𝑈) → ((𝐺𝑟) ⊆ 𝑊𝑟 ⊆ (𝐺𝑊)))
1514rexbidva 3255 . . 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 5917 . . . . . 6 (𝐺𝑊) ⊆ dom 𝐺
2019, 11sseqtri 3951 . . . . 5 (𝐺𝑊) ⊆ (𝑋 × 𝑋)
2120a1i 11 . . . 4 ((𝜑𝑟𝑈) → (𝐺𝑊) ⊆ (𝑋 × 𝑋))
22 ustssel 22821 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟𝑈 ∧ (𝐺𝑊) ⊆ (𝑋 × 𝑋)) → (𝑟 ⊆ (𝐺𝑊) → (𝐺𝑊) ∈ 𝑈))
2317, 18, 21, 22syl3anc 1368 . . 3 ((𝜑𝑟𝑈) → (𝑟 ⊆ (𝐺𝑊) → (𝐺𝑊) ∈ 𝑈))
2423rexlimdva 3243 . 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 1538   ∈ wcel 2111  ∃wrex 3107   ⊆ wss 3881  ⟨cop 4531   × cxp 5518  ◡ccnv 5519  dom cdm 5520   “ cima 5523  Fun wfun 6319  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138  UnifOncust 22815   Cnucucn 22891 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675  df-map 8394  df-ust 22816  df-ucn 22892 This theorem is referenced by:  fmucnd  22908
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