imasf1obl - Metamath Proof Explorer

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Theorem imasf1obl 24383

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 7255	. . . . . . . . . 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 7406	. . . . . . . 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 6815	. . . . . . . . . . . 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 6803	. . . . . . . . . . 11 ⊢ (^◡𝐹:𝐵–1-1-onto→𝑉 → ^◡𝐹:𝐵⟶𝑉)
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ^◡𝐹:𝐵⟶𝑉)
22	21	ffvelcdmda 7059	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (^◡𝐹‘𝑥) ∈ 𝑉)
23	6, 8, 9, 11, 12, 13, 15, 17, 22	imasdsf1o 24269	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑃)𝐷(𝐹‘(^◡𝐹‘𝑥))) = (𝑃𝐸(^◡𝐹‘𝑥)))
24	4, 23	eqtr3d 2767	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑃)𝐷𝑥) = (𝑃𝐸(^◡𝐹‘𝑥)))
25	24	breq1d 5120	. . . . . 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 24285	. . . . . . 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 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 24270	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))
33		f1of 6803	. . . . . . 7 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉⟶𝐵)
34	1, 33	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
35	34, 16	ffvelcdmd 7060	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐵)
36		elbl 24283	. . . . 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 1373	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝐵 ∧ ((𝐹‘𝑃)𝐷𝑥) < 𝑆)))
38		f1ofn 6804	. . . . 5 ⊢ (^◡𝐹:𝐵–1-1-onto→𝑉 → ^◡𝐹 Fn 𝐵)
39		elpreima 7033	. . . . 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 2728	. 2 ⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (^◡^◡𝐹 “ (𝑃(ball‘𝐸)𝑆)))
43		imacnvcnv 6182	. 2 ⊢ (^◡^◡𝐹 “ (𝑃(ball‘𝐸)𝑆)) = (𝐹 “ (𝑃(ball‘𝐸)𝑆))
44	42, 43	eqtrdi 2781	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 1540 ∈ wcel 2109 class class class wbr 5110 × cxp 5639 ^◡ccnv 5640 ↾ cres 5643 “ cima 5644 Fn wfn 6509 ⟶wf 6510 –1-1-onto→wf1o 6513 ‘cfv 6514 (class class class)co 7390 ℝ^*cxr 11214 < clt 11215 Basecbs 17186 distcds 17236 “_s cimas 17474 ∞Metcxmet 21256 ballcbl 21258

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-fz 13476 df-fzo 13623 df-seq 13974 df-hash 14303 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-0g 17411 df-gsum 17412 df-xrs 17472 df-imas 17478 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-bl 21266

This theorem is referenced by: imasf1oxms 24384

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