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Theorem ucnprima 24108
Description: The preimage by a uniformly continuous function 𝐹 of an entourage π‘Š of π‘Œ 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 (πœ‘ β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
ucnprima.2 (πœ‘ β†’ 𝑉 ∈ (UnifOnβ€˜π‘Œ))
ucnprima.3 (πœ‘ β†’ 𝐹 ∈ (π‘ˆ Cnu𝑉))
ucnprima.4 (πœ‘ β†’ π‘Š ∈ 𝑉)
ucnprima.5 𝐺 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(πΉβ€˜π‘₯), (πΉβ€˜π‘¦)⟩)
Assertion
Ref Expression
ucnprima (πœ‘ β†’ (◑𝐺 β€œ π‘Š) ∈ π‘ˆ)
Distinct variable groups:   π‘₯,𝑦,𝐹   π‘₯,𝑋,𝑦   π‘₯,𝐺,𝑦   π‘₯,π‘ˆ,𝑦   π‘₯,𝑉   π‘₯,π‘Š,𝑦   π‘₯,π‘Œ   πœ‘,π‘₯,𝑦
Allowed substitution hints:   𝑉(𝑦)   π‘Œ(𝑦)
Proof of Theorem ucnprima
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 ucnprima.1 . . . 4 (πœ‘ β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
2 ucnprima.2 . . . 4 (πœ‘ β†’ 𝑉 ∈ (UnifOnβ€˜π‘Œ))
3 ucnprima.3 . . . 4 (πœ‘ β†’ 𝐹 ∈ (π‘ˆ Cnu𝑉))
4 ucnprima.4 . . . 4 (πœ‘ β†’ π‘Š ∈ 𝑉)
5 ucnprima.5 . . . 4 𝐺 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(πΉβ€˜π‘₯), (πΉβ€˜π‘¦)⟩)
61, 2, 3, 4, 5ucnima 24107 . . 3 (πœ‘ β†’ βˆƒπ‘Ÿ ∈ π‘ˆ (𝐺 β€œ π‘Ÿ) βŠ† π‘Š)
75mpofun 7524 . . . . 5 Fun 𝐺
8 ustssxp 24030 . . . . . . 7 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ÿ ∈ π‘ˆ) β†’ π‘Ÿ βŠ† (𝑋 Γ— 𝑋))
91, 8sylan 579 . . . . . 6 ((πœ‘ ∧ π‘Ÿ ∈ π‘ˆ) β†’ π‘Ÿ βŠ† (𝑋 Γ— 𝑋))
10 opex 5454 . . . . . . 7 ⟨(πΉβ€˜π‘₯), (πΉβ€˜π‘¦)⟩ ∈ V
115, 10dmmpo 8050 . . . . . 6 dom 𝐺 = (𝑋 Γ— 𝑋)
129, 11sseqtrrdi 4025 . . . . 5 ((πœ‘ ∧ π‘Ÿ ∈ π‘ˆ) β†’ π‘Ÿ βŠ† dom 𝐺)
13 funimass3 7045 . . . . 5 ((Fun 𝐺 ∧ π‘Ÿ βŠ† dom 𝐺) β†’ ((𝐺 β€œ π‘Ÿ) βŠ† π‘Š ↔ π‘Ÿ βŠ† (◑𝐺 β€œ π‘Š)))
147, 12, 13sylancr 586 . . . 4 ((πœ‘ ∧ π‘Ÿ ∈ π‘ˆ) β†’ ((𝐺 β€œ π‘Ÿ) βŠ† π‘Š ↔ π‘Ÿ βŠ† (◑𝐺 β€œ π‘Š)))
1514rexbidva 3168 . . 3 (πœ‘ β†’ (βˆƒπ‘Ÿ ∈ π‘ˆ (𝐺 β€œ π‘Ÿ) βŠ† π‘Š ↔ βˆƒπ‘Ÿ ∈ π‘ˆ π‘Ÿ βŠ† (◑𝐺 β€œ π‘Š)))
166, 15mpbid 231 . 2 (πœ‘ β†’ βˆƒπ‘Ÿ ∈ π‘ˆ π‘Ÿ βŠ† (◑𝐺 β€œ π‘Š))
171adantr 480 . . . 4 ((πœ‘ ∧ π‘Ÿ ∈ π‘ˆ) β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
18 simpr 484 . . . 4 ((πœ‘ ∧ π‘Ÿ ∈ π‘ˆ) β†’ π‘Ÿ ∈ π‘ˆ)
19 cnvimass 6070 . . . . . 6 (◑𝐺 β€œ π‘Š) βŠ† dom 𝐺
2019, 11sseqtri 4010 . . . . 5 (◑𝐺 β€œ π‘Š) βŠ† (𝑋 Γ— 𝑋)
2120a1i 11 . . . 4 ((πœ‘ ∧ π‘Ÿ ∈ π‘ˆ) β†’ (◑𝐺 β€œ π‘Š) βŠ† (𝑋 Γ— 𝑋))
22 ustssel 24031 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ π‘Ÿ ∈ π‘ˆ ∧ (◑𝐺 β€œ π‘Š) βŠ† (𝑋 Γ— 𝑋)) β†’ (π‘Ÿ βŠ† (◑𝐺 β€œ π‘Š) β†’ (◑𝐺 β€œ π‘Š) ∈ π‘ˆ))
2317, 18, 21, 22syl3anc 1368 . . 3 ((πœ‘ ∧ π‘Ÿ ∈ π‘ˆ) β†’ (π‘Ÿ βŠ† (◑𝐺 β€œ π‘Š) β†’ (◑𝐺 β€œ π‘Š) ∈ π‘ˆ))
2423rexlimdva 3147 . 2 (πœ‘ β†’ (βˆƒπ‘Ÿ ∈ π‘ˆ π‘Ÿ βŠ† (◑𝐺 β€œ π‘Š) β†’ (◑𝐺 β€œ π‘Š) ∈ π‘ˆ))
2516, 24mpd 15 1 (πœ‘ β†’ (◑𝐺 β€œ π‘Š) ∈ π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3062   βŠ† wss 3940  βŸ¨cop 4626   Γ— cxp 5664  β—‘ccnv 5665  dom cdm 5666   β€œ cima 5669  Fun wfun 6527  β€˜cfv 6533  (class class class)co 7401   ∈ cmpo 7403  UnifOncust 24025   Cnucucn 24101
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-map 8817  df-ust 24026  df-ucn 24102
This theorem is referenced by:  fmucnd  24118
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