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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 22893 | . . 3 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊) |
7 | 5 | mpofun 7279 | . . . . 5 ⊢ Fun 𝐺 |
8 | ustssxp 22816 | . . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋)) | |
9 | 1, 8 | sylan 582 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ (𝑋 × 𝑋)) |
10 | opex 5359 | . . . . . . 7 ⊢ 〈(𝐹‘𝑥), (𝐹‘𝑦)〉 ∈ V | |
11 | 5, 10 | dmmpo 7772 | . . . . . 6 ⊢ dom 𝐺 = (𝑋 × 𝑋) |
12 | 9, 11 | sseqtrrdi 4021 | . . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ⊆ dom 𝐺) |
13 | funimass3 6827 | . . . . 5 ⊢ ((Fun 𝐺 ∧ 𝑟 ⊆ dom 𝐺) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ 𝑟 ⊆ (◡𝐺 “ 𝑊))) | |
14 | 7, 12, 13 | sylancr 589 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → ((𝐺 “ 𝑟) ⊆ 𝑊 ↔ 𝑟 ⊆ (◡𝐺 “ 𝑊))) |
15 | 14 | rexbidva 3299 | . . 3 ⊢ (𝜑 → (∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊 ↔ ∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊))) |
16 | 6, 15 | mpbid 234 | . 2 ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 𝑟 ⊆ (◡𝐺 “ 𝑊)) |
17 | 1 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋)) |
18 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → 𝑟 ∈ 𝑈) | |
19 | cnvimass 5952 | . . . . . 6 ⊢ (◡𝐺 “ 𝑊) ⊆ dom 𝐺 | |
20 | 19, 11 | sseqtri 4006 | . . . . 5 ⊢ (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋) |
21 | 20 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋)) |
22 | ustssel 22817 | . . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑟 ∈ 𝑈 ∧ (◡𝐺 “ 𝑊) ⊆ (𝑋 × 𝑋)) → (𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈)) | |
23 | 17, 18, 21, 22 | syl3anc 1367 | . . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑈) → (𝑟 ⊆ (◡𝐺 “ 𝑊) → (◡𝐺 “ 𝑊) ∈ 𝑈)) |
24 | 23 | rexlimdva 3287 | . 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 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 ⊆ wss 3939 〈cop 4576 × cxp 5556 ◡ccnv 5557 dom cdm 5558 “ cima 5561 Fun wfun 6352 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 UnifOncust 22811 Cnucucn 22887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-map 8411 df-ust 22812 df-ucn 22888 |
This theorem is referenced by: fmucnd 22904 |
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