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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 7223 | . . . . . . . . . 10 ⊢ ((𝐹:𝑉–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥) | |
3 | 1, 2 | sylan 580 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥) |
4 | 3 | oveq2d 7373 | . . . . . . . 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 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑈 = (𝐹 “s 𝑅)) |
7 | imasf1obl.v | . . . . . . . . . 10 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
8 | 7 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑉 = (Base‘𝑅)) |
9 | 1 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐹:𝑉–1-1-onto→𝐵) |
10 | imasf1obl.r | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
11 | 10 | adantr 481 | . . . . . . . . 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 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐸 ∈ (∞Met‘𝑉)) |
16 | imasf1obl.x | . . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝑉) | |
17 | 16 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑃 ∈ 𝑉) |
18 | f1ocnv 6796 | . . . . . . . . . . . 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 6784 | . . . . . . . . . . 11 ⊢ (◡𝐹:𝐵–1-1-onto→𝑉 → ◡𝐹:𝐵⟶𝑉) | |
21 | 19, 20 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → ◡𝐹:𝐵⟶𝑉) |
22 | 21 | ffvelcdmda 7035 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘𝑥) ∈ 𝑉) |
23 | 6, 8, 9, 11, 12, 13, 15, 17, 22 | imasdsf1o 23727 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑃)𝐷(𝐹‘(◡𝐹‘𝑥))) = (𝑃𝐸(◡𝐹‘𝑥))) |
24 | 4, 23 | eqtr3d 2778 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑃)𝐷𝑥) = (𝑃𝐸(◡𝐹‘𝑥))) |
25 | 24 | breq1d 5115 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((𝐹‘𝑃)𝐷𝑥) < 𝑆 ↔ (𝑃𝐸(◡𝐹‘𝑥)) < 𝑆)) |
26 | imasf1obl.s | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ*) | |
27 | 26 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑆 ∈ ℝ*) |
28 | elbl2 23743 | . . . . . . 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 837 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘𝑥) ∈ (𝑃(ball‘𝐸)𝑆) ↔ (𝑃𝐸(◡𝐹‘𝑥)) < 𝑆)) |
30 | 25, 29 | bitr4d 281 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((𝐹‘𝑃)𝐷𝑥) < 𝑆 ↔ (◡𝐹‘𝑥) ∈ (𝑃(ball‘𝐸)𝑆))) |
31 | 30 | pm5.32da 579 | . . . 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 23728 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
33 | f1of 6784 | . . . . . . 7 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉⟶𝐵) | |
34 | 1, 33 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵) |
35 | 34, 16 | ffvelcdmd 7036 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐵) |
36 | elbl 23741 | . . . . 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 1371 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝐵 ∧ ((𝐹‘𝑃)𝐷𝑥) < 𝑆))) |
38 | f1ofn 6785 | . . . . 5 ⊢ (◡𝐹:𝐵–1-1-onto→𝑉 → ◡𝐹 Fn 𝐵) | |
39 | elpreima 7008 | . . . . 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 310 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑆) ↔ 𝑥 ∈ (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆)))) |
42 | 41 | eqrdv 2734 | . 2 ⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆))) |
43 | imacnvcnv 6158 | . 2 ⊢ (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆)) = (𝐹 “ (𝑃(ball‘𝐸)𝑆)) | |
44 | 42, 43 | eqtrdi 2792 | 1 ⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (𝐹 “ (𝑃(ball‘𝐸)𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5105 × cxp 5631 ◡ccnv 5632 ↾ cres 5635 “ cima 5636 Fn wfn 6491 ⟶wf 6492 –1-1-onto→wf1o 6495 ‘cfv 6496 (class class class)co 7357 ℝ*cxr 11188 < clt 11189 Basecbs 17083 distcds 17142 “s cimas 17386 ∞Metcxmet 20781 ballcbl 20783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-0g 17323 df-gsum 17324 df-xrs 17384 df-imas 17390 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-bl 20791 |
This theorem is referenced by: imasf1oxms 23845 |
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