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Theorem imasf1obl 24522

Description: The image of a metric space ball. (Contributed by Mario Carneiro, 28-Aug-2015.)

**Hypotheses**
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^*)

**Assertion**
Ref	Expression
imasf1obl	⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (𝐹 “ (𝑃(ball‘𝐸)𝑆)))

Proof of Theorem imasf1obl Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasf1obl.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)
2		f1ocnvfv2 7313	. . . . . . . . . 10 ⊢ ((𝐹:𝑉–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
3	1, 2	sylan 579	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
4	3	oveq2d 7464	. . . . . . . 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 6874	. . . . . . . . . . . 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 6862	. . . . . . . . . . 11 ⊢ (^◡𝐹:𝐵–1-1-onto→𝑉 → ^◡𝐹:𝐵⟶𝑉)
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ^◡𝐹:𝐵⟶𝑉)
22	21	ffvelcdmda 7118	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (^◡𝐹‘𝑥) ∈ 𝑉)
23	6, 8, 9, 11, 12, 13, 15, 17, 22	imasdsf1o 24405	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑃)𝐷(𝐹‘(^◡𝐹‘𝑥))) = (𝑃𝐸(^◡𝐹‘𝑥)))
24	4, 23	eqtr3d 2782	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑃)𝐷𝑥) = (𝑃𝐸(^◡𝐹‘𝑥)))
25	24	breq1d 5176	. . . . . 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 24421	. . . . . . 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 838	. . . . . 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 578	. . . 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 24406	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))
33		f1of 6862	. . . . . . 7 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉⟶𝐵)
34	1, 33	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
35	34, 16	ffvelcdmd 7119	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐵)
36		elbl 24419	. . . . 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 6863	. . . . 5 ⊢ (^◡𝐹:𝐵–1-1-onto→𝑉 → ^◡𝐹 Fn 𝐵)
39		elpreima 7091	. . . . 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 2738	. 2 ⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (^◡^◡𝐹 “ (𝑃(ball‘𝐸)𝑆)))
43		imacnvcnv 6237	. 2 ⊢ (^◡^◡𝐹 “ (𝑃(ball‘𝐸)𝑆)) = (𝐹 “ (𝑃(ball‘𝐸)𝑆))
44	42, 43	eqtrdi 2796	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 2108 class class class wbr 5166 × cxp 5698 ^◡ccnv 5699 ↾ cres 5702 “ cima 5703 Fn wfn 6568 ⟶wf 6569 –1-1-onto→wf1o 6572 ‘cfv 6573 (class class class)co 7448 ℝ^*cxr 11323 < clt 11324 Basecbs 17258 distcds 17320 “_s cimas 17564 ∞Metcxmet 21372 ballcbl 21374

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-0g 17501 df-gsum 17502 df-xrs 17562 df-imas 17568 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-bl 21382

This theorem is referenced by: imasf1oxms 24523

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