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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 7297 | . . . . . . . . . 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 7447 | . . . . . . . 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 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑈 = (𝐹 “s 𝑅)) |
7 | imasf1obl.v | . . . . . . . . . 10 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
8 | 7 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑉 = (Base‘𝑅)) |
9 | 1 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐹:𝑉–1-1-onto→𝐵) |
10 | imasf1obl.r | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
11 | 10 | adantr 480 | . . . . . . . . 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 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐸 ∈ (∞Met‘𝑉)) |
16 | imasf1obl.x | . . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝑉) | |
17 | 16 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑃 ∈ 𝑉) |
18 | f1ocnv 6861 | . . . . . . . . . . . 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 6849 | . . . . . . . . . . 11 ⊢ (◡𝐹:𝐵–1-1-onto→𝑉 → ◡𝐹:𝐵⟶𝑉) | |
21 | 19, 20 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → ◡𝐹:𝐵⟶𝑉) |
22 | 21 | ffvelcdmda 7104 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘𝑥) ∈ 𝑉) |
23 | 6, 8, 9, 11, 12, 13, 15, 17, 22 | imasdsf1o 24400 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑃)𝐷(𝐹‘(◡𝐹‘𝑥))) = (𝑃𝐸(◡𝐹‘𝑥))) |
24 | 4, 23 | eqtr3d 2777 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑃)𝐷𝑥) = (𝑃𝐸(◡𝐹‘𝑥))) |
25 | 24 | breq1d 5158 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((𝐹‘𝑃)𝐷𝑥) < 𝑆 ↔ (𝑃𝐸(◡𝐹‘𝑥)) < 𝑆)) |
26 | imasf1obl.s | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ*) | |
27 | 26 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑆 ∈ ℝ*) |
28 | elbl2 24416 | . . . . . . 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 839 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘𝑥) ∈ (𝑃(ball‘𝐸)𝑆) ↔ (𝑃𝐸(◡𝐹‘𝑥)) < 𝑆)) |
30 | 25, 29 | bitr4d 282 | . . . . 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 24401 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
33 | f1of 6849 | . . . . . . 7 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉⟶𝐵) | |
34 | 1, 33 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵) |
35 | 34, 16 | ffvelcdmd 7105 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐵) |
36 | elbl 24414 | . . . . 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 1370 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝐵 ∧ ((𝐹‘𝑃)𝐷𝑥) < 𝑆))) |
38 | f1ofn 6850 | . . . . 5 ⊢ (◡𝐹:𝐵–1-1-onto→𝑉 → ◡𝐹 Fn 𝐵) | |
39 | elpreima 7078 | . . . . 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 311 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑆) ↔ 𝑥 ∈ (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆)))) |
42 | 41 | eqrdv 2733 | . 2 ⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆))) |
43 | imacnvcnv 6228 | . 2 ⊢ (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆)) = (𝐹 “ (𝑃(ball‘𝐸)𝑆)) | |
44 | 42, 43 | eqtrdi 2791 | 1 ⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (𝐹 “ (𝑃(ball‘𝐸)𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 × cxp 5687 ◡ccnv 5688 ↾ cres 5691 “ cima 5692 Fn wfn 6558 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 ℝ*cxr 11292 < clt 11293 Basecbs 17245 distcds 17307 “s cimas 17551 ∞Metcxmet 21367 ballcbl 21369 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-0g 17488 df-gsum 17489 df-xrs 17549 df-imas 17555 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-bl 21377 |
This theorem is referenced by: imasf1oxms 24518 |
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