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Theorem imasf1obl 23750

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 7205	. . . . . . . . . 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 7353	. . . . . . . 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 6779	. . . . . . . . . . . 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 6767	. . . . . . . . . . 11 ⊢ (^◡𝐹:𝐵–1-1-onto→𝑉 → ^◡𝐹:𝐵⟶𝑉)
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ^◡𝐹:𝐵⟶𝑉)
22	21	ffvelcdmda 7017	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (^◡𝐹‘𝑥) ∈ 𝑉)
23	6, 8, 9, 11, 12, 13, 15, 17, 22	imasdsf1o 23633	. . . . . . . 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 5102	. . . . . 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 23649	. . . . . . 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 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 23634	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))
33		f1of 6767	. . . . . . 7 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉⟶𝐵)
34	1, 33	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵)
35	34, 16	ffvelcdmd 7018	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐵)
36		elbl 23647	. . . . 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 6768	. . . . 5 ⊢ (^◡𝐹:𝐵–1-1-onto→𝑉 → ^◡𝐹 Fn 𝐵)
39		elpreima 6991	. . . . 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 6144	. 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 1540 ∈ wcel 2105 class class class wbr 5092 × cxp 5618 ^◡ccnv 5619 ↾ cres 5622 “ cima 5623 Fn wfn 6474 ⟶wf 6475 –1-1-onto→wf1o 6478 ‘cfv 6479 (class class class)co 7337 ℝ^*cxr 11109 < clt 11110 Basecbs 17009 distcds 17068 “_s cimas 17312 ∞Metcxmet 20688 ballcbl 20690

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-pre-sup 11050

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-se 5576 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-isom 6488 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-of 7595 df-om 7781 df-1st 7899 df-2nd 7900 df-supp 8048 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-fsupp 9227 df-sup 9299 df-inf 9300 df-oi 9367 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-5 12140 df-6 12141 df-7 12142 df-8 12143 df-9 12144 df-n0 12335 df-z 12421 df-dec 12539 df-uz 12684 df-rp 12832 df-xneg 12949 df-xadd 12950 df-xmul 12951 df-fz 13341 df-fzo 13484 df-seq 13823 df-hash 14146 df-struct 16945 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulr 17073 df-sca 17075 df-vsca 17076 df-ip 17077 df-tset 17078 df-ple 17079 df-ds 17081 df-0g 17249 df-gsum 17250 df-xrs 17310 df-imas 17316 df-mre 17392 df-mrc 17393 df-acs 17395 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-submnd 18528 df-mulg 18797 df-cntz 19019 df-cmn 19483 df-psmet 20695 df-xmet 20696 df-bl 20698

This theorem is referenced by: imasf1oxms 23751

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