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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 22085 | . . 3 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊) |
7 | 5 | mpt2fun 6762 | . . . . 5 ⊢ Fun 𝐺 |
8 | ustssxp 22008 | . . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋)) | |
9 | 1, 8 | sylan 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋)) |
10 | opex 4932 | . . . . . . 7 ⊢ 〈(𝐹‘𝑥), (𝐹‘𝑦)〉 ∈ V | |
11 | 5, 10 | dmmpt2 7240 | . . . . . 6 ⊢ dom 𝐺 = (𝑋 × 𝑋) |
12 | 9, 11 | syl6sseqr 3652 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ dom 𝐺) |
13 | funimass3 6333 | . . . . 5 ⊢ ((Fun 𝐺 ∧ 𝑟 ⊆ dom 𝐺) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ 𝑟 ⊆ (◡𝐺 “ 𝑊))) | |
14 | 7, 12, 13 | sylancr 695 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ 𝑟 ⊆ (◡𝐺 “ 𝑊))) |
15 | 14 | rexbidva 3049 | . . 3 ⊢ (𝜑 → (∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊 ↔ ∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊))) |
16 | 6, 15 | mpbid 222 | . 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊)) |
17 | 1 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋)) |
18 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ∈ 𝑈) | |
19 | cnvimass 5485 | . . . . . 6 ⊢ (◡𝐺 “ 𝑊) ⊆ dom 𝐺 | |
20 | 19, 11 | sseqtri 3637 | . . . . 5 ⊢ (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋) |
21 | 20 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋)) |
22 | ustssel 22009 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈 ∧ (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋)) → (𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈)) | |
23 | 17, 18, 21, 22 | syl3anc 1326 | . . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈)) |
24 | 23 | rexlimdva 3031 | . 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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ⊆ wss 3574 〈cop 4183 × cxp 5112 ◡ccnv 5113 dom cdm 5114 “ cima 5117 Fun wfun 5882 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 UnifOncust 22003 Cnucucn 22079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-ust 22004 df-ucn 22080 |
This theorem is referenced by: fmucnd 22096 |
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