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Theorem ucnprima 22414
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 22413 . . 3 (𝜑 → ∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊)
75mpt2fun 6996 . . . . 5 Fun 𝐺
8 ustssxp 22336 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
91, 8sylan 576 . . . . . 6 ((𝜑𝑟𝑈) → 𝑟 ⊆ (𝑋 × 𝑋))
10 opex 5123 . . . . . . 7 ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ V
115, 10dmmpt2 7476 . . . . . 6 dom 𝐺 = (𝑋 × 𝑋)
129, 11syl6sseqr 3848 . . . . 5 ((𝜑𝑟𝑈) → 𝑟 ⊆ dom 𝐺)
13 funimass3 6559 . . . . 5 ((Fun 𝐺𝑟 ⊆ dom 𝐺) → ((𝐺𝑟) ⊆ 𝑊𝑟 ⊆ (𝐺𝑊)))
147, 12, 13sylancr 582 . . . 4 ((𝜑𝑟𝑈) → ((𝐺𝑟) ⊆ 𝑊𝑟 ⊆ (𝐺𝑊)))
1514rexbidva 3230 . . 3 (𝜑 → (∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊 ↔ ∃𝑟𝑈 𝑟 ⊆ (𝐺𝑊)))
166, 15mpbid 224 . 2 (𝜑 → ∃𝑟𝑈 𝑟 ⊆ (𝐺𝑊))
171adantr 473 . . . 4 ((𝜑𝑟𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
18 simpr 478 . . . 4 ((𝜑𝑟𝑈) → 𝑟𝑈)
19 cnvimass 5702 . . . . . 6 (𝐺𝑊) ⊆ dom 𝐺
2019, 11sseqtri 3833 . . . . 5 (𝐺𝑊) ⊆ (𝑋 × 𝑋)
2120a1i 11 . . . 4 ((𝜑𝑟𝑈) → (𝐺𝑊) ⊆ (𝑋 × 𝑋))
22 ustssel 22337 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟𝑈 ∧ (𝐺𝑊) ⊆ (𝑋 × 𝑋)) → (𝑟 ⊆ (𝐺𝑊) → (𝐺𝑊) ∈ 𝑈))
2317, 18, 21, 22syl3anc 1491 . . 3 ((𝜑𝑟𝑈) → (𝑟 ⊆ (𝐺𝑊) → (𝐺𝑊) ∈ 𝑈))
2423rexlimdva 3212 . 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 198  wa 385   = wceq 1653  wcel 2157  wrex 3090  wss 3769  cop 4374   × cxp 5310  ccnv 5311  dom cdm 5312  cima 5315  Fun wfun 6095  cfv 6101  (class class class)co 6878  cmpt2 6880  UnifOncust 22331   Cnucucn 22407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-map 8097  df-ust 22332  df-ucn 22408
This theorem is referenced by:  fmucnd  22424
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