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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 21895 | . . 3 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊) |
7 | 5 | mpt2fun 6660 | . . . . 5 ⊢ Fun 𝐺 |
8 | ustssxp 21818 | . . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋)) | |
9 | 1, 8 | sylan 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋)) |
10 | opex 4859 | . . . . . . 7 ⊢ 〈(𝐹‘𝑥), (𝐹‘𝑦)〉 ∈ V | |
11 | 5, 10 | dmmpt2 7129 | . . . . . 6 ⊢ dom 𝐺 = (𝑋 × 𝑋) |
12 | 9, 11 | syl6sseqr 3615 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ dom 𝐺) |
13 | funimass3 6241 | . . . . 5 ⊢ ((Fun 𝐺 ∧ 𝑟 ⊆ dom 𝐺) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ 𝑟 ⊆ (◡𝐺 “ 𝑊))) | |
14 | 7, 12, 13 | sylancr 694 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ 𝑟 ⊆ (◡𝐺 “ 𝑊))) |
15 | 14 | rexbidva 3031 | . . 3 ⊢ (𝜑 → (∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊 ↔ ∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊))) |
16 | 6, 15 | mpbid 221 | . 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊)) |
17 | 1 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋)) |
18 | simpr 476 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ∈ 𝑈) | |
19 | cnvimass 5404 | . . . . . 6 ⊢ (◡𝐺 “ 𝑊) ⊆ dom 𝐺 | |
20 | 19, 11 | sseqtri 3600 | . . . . 5 ⊢ (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋) |
21 | 20 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋)) |
22 | ustssel 21819 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈 ∧ (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋)) → (𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈)) | |
23 | 17, 18, 21, 22 | syl3anc 1318 | . . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈)) |
24 | 23 | rexlimdva 3013 | . 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ⊆ wss 3540 〈cop 4131 × cxp 5036 ◡ccnv 5037 dom cdm 5038 “ cima 5041 Fun wfun 5798 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 UnifOncust 21813 Cnucucn 21889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-ust 21814 df-ucn 21890 |
This theorem is referenced by: fmucnd 21906 |
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