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Mirrors > Home > MPE Home > Th. List > imasf1obl | Structured version Visualization version GIF version |
Description: The image of a metric space ball. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
imasf1obl.u | ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
imasf1obl.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
imasf1obl.f | ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) |
imasf1obl.r | ⊢ (𝜑 → 𝑅 ∈ 𝑍) |
imasf1obl.e | ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) |
imasf1obl.d | ⊢ 𝐷 = (dist‘𝑈) |
imasf1obl.m | ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉)) |
imasf1obl.x | ⊢ (𝜑 → 𝑃 ∈ 𝑉) |
imasf1obl.s | ⊢ (𝜑 → 𝑆 ∈ ℝ*) |
Ref | Expression |
---|---|
imasf1obl | ⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (𝐹 “ (𝑃(ball‘𝐸)𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasf1obl.f | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) | |
2 | f1ocnvfv2 7033 | . . . . . . . . . 10 ⊢ ((𝐹:𝑉–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥) | |
3 | 1, 2 | sylan 582 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥) |
4 | 3 | oveq2d 7171 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑃)𝐷(𝐹‘(◡𝐹‘𝑥))) = ((𝐹‘𝑃)𝐷𝑥)) |
5 | imasf1obl.u | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | |
6 | 5 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑈 = (𝐹 “s 𝑅)) |
7 | imasf1obl.v | . . . . . . . . . 10 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
8 | 7 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑉 = (Base‘𝑅)) |
9 | 1 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐹:𝑉–1-1-onto→𝐵) |
10 | imasf1obl.r | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
11 | 10 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ 𝑍) |
12 | imasf1obl.e | . . . . . . . . 9 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) | |
13 | imasf1obl.d | . . . . . . . . 9 ⊢ 𝐷 = (dist‘𝑈) | |
14 | imasf1obl.m | . . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉)) | |
15 | 14 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐸 ∈ (∞Met‘𝑉)) |
16 | imasf1obl.x | . . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝑉) | |
17 | 16 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑃 ∈ 𝑉) |
18 | f1ocnv 6626 | . . . . . . . . . . . 12 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝑉) | |
19 | 1, 18 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → ◡𝐹:𝐵–1-1-onto→𝑉) |
20 | f1of 6614 | . . . . . . . . . . 11 ⊢ (◡𝐹:𝐵–1-1-onto→𝑉 → ◡𝐹:𝐵⟶𝑉) | |
21 | 19, 20 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → ◡𝐹:𝐵⟶𝑉) |
22 | 21 | ffvelrnda 6850 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘𝑥) ∈ 𝑉) |
23 | 6, 8, 9, 11, 12, 13, 15, 17, 22 | imasdsf1o 22983 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑃)𝐷(𝐹‘(◡𝐹‘𝑥))) = (𝑃𝐸(◡𝐹‘𝑥))) |
24 | 4, 23 | eqtr3d 2858 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑃)𝐷𝑥) = (𝑃𝐸(◡𝐹‘𝑥))) |
25 | 24 | breq1d 5075 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((𝐹‘𝑃)𝐷𝑥) < 𝑆 ↔ (𝑃𝐸(◡𝐹‘𝑥)) < 𝑆)) |
26 | imasf1obl.s | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ*) | |
27 | 26 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑆 ∈ ℝ*) |
28 | elbl2 22999 | . . . . . . 7 ⊢ (((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑆 ∈ ℝ*) ∧ (𝑃 ∈ 𝑉 ∧ (◡𝐹‘𝑥) ∈ 𝑉)) → ((◡𝐹‘𝑥) ∈ (𝑃(ball‘𝐸)𝑆) ↔ (𝑃𝐸(◡𝐹‘𝑥)) < 𝑆)) | |
29 | 15, 27, 17, 22, 28 | syl22anc 836 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘𝑥) ∈ (𝑃(ball‘𝐸)𝑆) ↔ (𝑃𝐸(◡𝐹‘𝑥)) < 𝑆)) |
30 | 25, 29 | bitr4d 284 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((𝐹‘𝑃)𝐷𝑥) < 𝑆 ↔ (◡𝐹‘𝑥) ∈ (𝑃(ball‘𝐸)𝑆))) |
31 | 30 | pm5.32da 581 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ ((𝐹‘𝑃)𝐷𝑥) < 𝑆) ↔ (𝑥 ∈ 𝐵 ∧ (◡𝐹‘𝑥) ∈ (𝑃(ball‘𝐸)𝑆)))) |
32 | 5, 7, 1, 10, 12, 13, 14 | imasf1oxmet 22984 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
33 | f1of 6614 | . . . . . . 7 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉⟶𝐵) | |
34 | 1, 33 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵) |
35 | 34, 16 | ffvelrnd 6851 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐵) |
36 | elbl 22997 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ (𝐹‘𝑃) ∈ 𝐵 ∧ 𝑆 ∈ ℝ*) → (𝑥 ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝐵 ∧ ((𝐹‘𝑃)𝐷𝑥) < 𝑆))) | |
37 | 32, 35, 26, 36 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝐵 ∧ ((𝐹‘𝑃)𝐷𝑥) < 𝑆))) |
38 | f1ofn 6615 | . . . . 5 ⊢ (◡𝐹:𝐵–1-1-onto→𝑉 → ◡𝐹 Fn 𝐵) | |
39 | elpreima 6827 | . . . . 5 ⊢ (◡𝐹 Fn 𝐵 → (𝑥 ∈ (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆)) ↔ (𝑥 ∈ 𝐵 ∧ (◡𝐹‘𝑥) ∈ (𝑃(ball‘𝐸)𝑆)))) | |
40 | 19, 38, 39 | 3syl 18 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆)) ↔ (𝑥 ∈ 𝐵 ∧ (◡𝐹‘𝑥) ∈ (𝑃(ball‘𝐸)𝑆)))) |
41 | 31, 37, 40 | 3bitr4d 313 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑆) ↔ 𝑥 ∈ (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆)))) |
42 | 41 | eqrdv 2819 | . 2 ⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆))) |
43 | imacnvcnv 6062 | . 2 ⊢ (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆)) = (𝐹 “ (𝑃(ball‘𝐸)𝑆)) | |
44 | 42, 43 | syl6eq 2872 | 1 ⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (𝐹 “ (𝑃(ball‘𝐸)𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 × cxp 5552 ◡ccnv 5553 ↾ cres 5556 “ cima 5557 Fn wfn 6349 ⟶wf 6350 –1-1-onto→wf1o 6353 ‘cfv 6354 (class class class)co 7155 ℝ*cxr 10673 < clt 10674 Basecbs 16482 distcds 16573 “s cimas 16776 ∞Metcxmet 20529 ballcbl 20531 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-0g 16714 df-gsum 16715 df-xrs 16774 df-imas 16780 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-bl 20539 |
This theorem is referenced by: imasf1oxms 23098 |
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