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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 7028 | . . . . . . . . . 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 7166 | . . . . . . . 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 6621 | . . . . . . . . . . . 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 6609 | . . . . . . . . . . 11 ⊢ (◡𝐹:𝐵–1-1-onto→𝑉 → ◡𝐹:𝐵⟶𝑉) | |
21 | 19, 20 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → ◡𝐹:𝐵⟶𝑉) |
22 | 21 | ffvelrnda 6845 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘𝑥) ∈ 𝑉) |
23 | 6, 8, 9, 11, 12, 13, 15, 17, 22 | imasdsf1o 22978 | . . . . . . . 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 5068 | . . . . . 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 22994 | . . . . . . 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 22979 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
33 | f1of 6609 | . . . . . . 7 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉⟶𝐵) | |
34 | 1, 33 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵) |
35 | 34, 16 | ffvelrnd 6846 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐵) |
36 | elbl 22992 | . . . . 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 6610 | . . . . 5 ⊢ (◡𝐹:𝐵–1-1-onto→𝑉 → ◡𝐹 Fn 𝐵) | |
39 | elpreima 6822 | . . . . 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 6057 | . 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 5058 × cxp 5547 ◡ccnv 5548 ↾ cres 5551 “ cima 5552 Fn wfn 6344 ⟶wf 6345 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7150 ℝ*cxr 10668 < clt 10669 Basecbs 16477 distcds 16568 “s cimas 16771 ∞Metcxmet 20524 ballcbl 20526 |
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 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-0g 16709 df-gsum 16710 df-xrs 16769 df-imas 16775 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-bl 20534 |
This theorem is referenced by: imasf1oxms 23093 |
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