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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 24306 | . . 3 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊) |
7 | 5 | mpofun 7557 | . . . . 5 ⊢ Fun 𝐺 |
8 | ustssxp 24229 | . . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋)) | |
9 | 1, 8 | sylan 580 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋)) |
10 | opex 5475 | . . . . . . 7 ⊢ 〈(𝐹‘𝑥), (𝐹‘𝑦)〉 ∈ V | |
11 | 5, 10 | dmmpo 8095 | . . . . . 6 ⊢ dom 𝐺 = (𝑋 × 𝑋) |
12 | 9, 11 | sseqtrrdi 4047 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ dom 𝐺) |
13 | funimass3 7074 | . . . . 5 ⊢ ((Fun 𝐺 ∧ 𝑟 ⊆ dom 𝐺) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ 𝑟 ⊆ (◡𝐺 “ 𝑊))) | |
14 | 7, 12, 13 | sylancr 587 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ 𝑟 ⊆ (◡𝐺 “ 𝑊))) |
15 | 14 | rexbidva 3175 | . . 3 ⊢ (𝜑 → (∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊 ↔ ∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊))) |
16 | 6, 15 | mpbid 232 | . 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊)) |
17 | 1 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋)) |
18 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ∈ 𝑈) | |
19 | cnvimass 6102 | . . . . . 6 ⊢ (◡𝐺 “ 𝑊) ⊆ dom 𝐺 | |
20 | 19, 11 | sseqtri 4032 | . . . . 5 ⊢ (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋) |
21 | 20 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋)) |
22 | ustssel 24230 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈 ∧ (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋)) → (𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈)) | |
23 | 17, 18, 21, 22 | syl3anc 1370 | . . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈)) |
24 | 23 | rexlimdva 3153 | . 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 〈cop 4637 × cxp 5687 ◡ccnv 5688 dom cdm 5689 “ cima 5692 Fun wfun 6557 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 UnifOncust 24224 Cnucucn 24300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-ust 24225 df-ucn 24301 |
This theorem is referenced by: fmucnd 24317 |
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