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Mirrors > Home > MPE Home > Th. List > ucnprima | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ucnprima.1 | ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋)) |
ucnprima.2 | ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌)) |
ucnprima.3 | ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉)) |
ucnprima.4 | ⊢ (𝜑 → 𝑊 ∈ 𝑉) |
ucnprima.5 | ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ 〈(𝐹‘𝑥), (𝐹‘𝑦)〉) |
Ref | Expression |
---|---|
ucnprima | ⊢ (𝜑 → (◡𝐺 “ 𝑊) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ucnprima.1 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋)) | |
2 | ucnprima.2 | . . . 4 ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌)) | |
3 | ucnprima.3 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉)) | |
4 | ucnprima.4 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑉) | |
5 | ucnprima.5 | . . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ 〈(𝐹‘𝑥), (𝐹‘𝑦)〉) | |
6 | 1, 2, 3, 4, 5 | ucnima 22887 | . . 3 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊) |
7 | 5 | mpofun 7255 | . . . . 5 ⊢ Fun 𝐺 |
8 | ustssxp 22810 | . . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋)) | |
9 | 1, 8 | sylan 583 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋)) |
10 | opex 5321 | . . . . . . 7 ⊢ 〈(𝐹‘𝑥), (𝐹‘𝑦)〉 ∈ V | |
11 | 5, 10 | dmmpo 7751 | . . . . . 6 ⊢ dom 𝐺 = (𝑋 × 𝑋) |
12 | 9, 11 | sseqtrrdi 3966 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ dom 𝐺) |
13 | funimass3 6801 | . . . . 5 ⊢ ((Fun 𝐺 ∧ 𝑟 ⊆ dom 𝐺) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ 𝑟 ⊆ (◡𝐺 “ 𝑊))) | |
14 | 7, 12, 13 | sylancr 590 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ 𝑟 ⊆ (◡𝐺 “ 𝑊))) |
15 | 14 | rexbidva 3255 | . . 3 ⊢ (𝜑 → (∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊 ↔ ∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊))) |
16 | 6, 15 | mpbid 235 | . 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊)) |
17 | 1 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋)) |
18 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ∈ 𝑈) | |
19 | cnvimass 5916 | . . . . . 6 ⊢ (◡𝐺 “ 𝑊) ⊆ dom 𝐺 | |
20 | 19, 11 | sseqtri 3951 | . . . . 5 ⊢ (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋) |
21 | 20 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋)) |
22 | ustssel 22811 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈 ∧ (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋)) → (𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈)) | |
23 | 17, 18, 21, 22 | syl3anc 1368 | . . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈)) |
24 | 23 | rexlimdva 3243 | . 2 ⊢ (𝜑 → (∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈)) |
25 | 16, 24 | mpd 15 | 1 ⊢ (𝜑 → (◡𝐺 “ 𝑊) ∈ 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 5517 ◡ccnv 5518 dom cdm 5519 “ cima 5522 Fun wfun 6318 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 UnifOncust 22805 Cnucucn 22881 |
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 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
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 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-ust 22806 df-ucn 22882 |
This theorem is referenced by: fmucnd 22898 |
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