Theorem imasf1oxms 18472

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

Hypotheses

Ref	Expression
imasf1obl.u	|- ( ph -> U = ( F "s R ) )
imasf1obl.v	|- ( ph -> V = ( Base ` R ) )
imasf1obl.f	|- ( ph -> F : V -1-1-onto-> B )
imasf1oxms.r	|- ( ph -> R e. * MetSp )

Assertion

Ref	Expression
imasf1oxms	|- ( ph -> U e. * MetSp )

Proof of Theorem imasf1oxms

Dummy variables x r y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasf1obl.u	. . . . 5 |- ( ph -> U = ( F "s R ) )
2		imasf1obl.v	. . . . 5 |- ( ph -> V = ( Base ` R ) )
3		imasf1obl.f	. . . . 5 |- ( ph -> F : V -1-1-onto-> B )
4		imasf1oxms.r	. . . . 5 |- ( ph -> R e. * MetSp )
5		eqid 2404	. . . . 5 |- ( ( dist ` R ) |` ( V X. V ) ) = ( ( dist ` R ) |` ( V X. V ) )
6		eqid 2404	. . . . 5 |- ( dist ` U ) = ( dist ` U )
7		eqid 2404	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
8		eqid 2404	. . . . . . . 8 |- ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) )
9	7, 8	xmsxmet 18439	. . . . . . 7 |- ( R e. * MetSp -> ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) e. ( * Met ` ( Base ` R ) ) )
10	4, 9	syl 16	. . . . . 6 |- ( ph -> ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) e. ( * Met ` ( Base ` R ) ) )
11	2, 2	xpeq12d 4862	. . . . . . 7 |- ( ph -> ( V X. V ) = ( ( Base ` R ) X. ( Base ` R ) ) )
12	11	reseq2d 5105	. . . . . 6 |- ( ph -> ( ( dist ` R ) |` ( V X. V ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) )
13	2	fveq2d 5691	. . . . . 6 |- ( ph -> ( * Met ` V ) = ( * Met ` ( Base ` R ) ) )
14	10, 12, 13	3eltr4d 2485	. . . . 5 |- ( ph -> ( ( dist ` R ) |` ( V X. V ) ) e. ( * Met ` V ) )
15	1, 2, 3, 4, 5, 6, 14	imasf1oxmet 18358	. . . 4 |- ( ph -> ( dist ` U ) e. ( * Met ` B ) )
16		f1ofo 5640	. . . . . . 7 |- ( F : V -1-1-onto-> B -> F : V -onto-> B )
17	3, 16	syl 16	. . . . . 6 |- ( ph -> F : V -onto-> B )
18	1, 2, 17, 4	imasbas 13693	. . . . 5 |- ( ph -> B = ( Base ` U ) )
19	18	fveq2d 5691	. . . 4 |- ( ph -> ( * Met ` B ) = ( * Met ` ( Base ` U ) ) )
20	15, 19	eleqtrd 2480	. . 3 |- ( ph -> ( dist ` U ) e. ( * Met ` ( Base ` U ) ) )
21		ssid 3327	. . 3 |- ( Base ` U ) C_ ( Base ` U )
22		xmetres2 18344	. . 3 |- ( ( ( dist ` U ) e. ( * Met ` ( Base ` U ) ) /\ ( Base ` U ) C_ ( Base ` U ) ) -> ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) e. ( * Met ` ( Base ` U ) ) )
23	20, 21, 22	sylancl 644	. 2 |- ( ph -> ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) e. ( * Met ` ( Base ` U ) ) )
24		eqid 2404	. . . 4 |- ( TopOpen ` R ) = ( TopOpen ` R )
25		eqid 2404	. . . 4 |- ( TopOpen ` U ) = ( TopOpen ` U )
26	1, 2, 17, 4, 24, 25	imastopn 17705	. . 3 |- ( ph -> ( TopOpen ` U ) = ( ( TopOpen ` R ) qTop F ) )
27	24, 7, 8	xmstopn 18434	. . . . . 6 |- ( R e. * MetSp -> ( TopOpen ` R ) = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) )
28	4, 27	syl 16	. . . . 5 |- ( ph -> ( TopOpen ` R ) = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) )
29	12	fveq2d 5691	. . . . 5 |- ( ph -> ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) )
30	28, 29	eqtr4d 2439	. . . 4 |- ( ph -> ( TopOpen ` R ) = ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) )
31	30	oveq1d 6055	. . 3 |- ( ph -> ( ( TopOpen ` R ) qTop F ) = ( ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) )
32		blbas 18413	. . . . . 6 |- ( ( ( dist ` R ) |` ( V X. V ) ) e. ( * Met ` V ) -> ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) e. TopBases )
33	14, 32	syl 16	. . . . 5 |- ( ph -> ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) e. TopBases )
34		unirnbl 18403	. . . . . . 7 |- ( ( ( dist ` R ) |` ( V X. V ) ) e. ( * Met ` V ) -> U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) = V )
35		f1oeq2 5625	. . . . . . 7 |- ( U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) = V -> ( F : U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) -1-1-onto-> B <-> F : V -1-1-onto-> B ) )
36	14, 34, 35	3syl 19	. . . . . 6 |- ( ph -> ( F : U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) -1-1-onto-> B <-> F : V -1-1-onto-> B ) )
37	3, 36	mpbird 224	. . . . 5 |- ( ph -> F : U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) -1-1-onto-> B )
38		eqid 2404	. . . . . 6 |- U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) = U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) )
39	38	tgqtop 17697	. . . . 5 |- ( ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) e. TopBases /\ F : U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) -1-1-onto-> B ) -> ( ( topGen ` ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) qTop F ) = ( topGen ` ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) ) )
40	33, 37, 39	syl2anc 643	. . . 4 |- ( ph -> ( ( topGen ` ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) qTop F ) = ( topGen ` ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) ) )
41		eqid 2404	. . . . . . 7 |- ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) = ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) )
42	41	mopnval 18421	. . . . . 6 |- ( ( ( dist ` R ) |` ( V X. V ) ) e. ( * Met ` V ) -> ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) = ( topGen ` ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) )
43	14, 42	syl 16	. . . . 5 |- ( ph -> ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) = ( topGen ` ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) )
44	43	oveq1d 6055	. . . 4 |- ( ph -> ( ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) = ( ( topGen ` ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) qTop F ) )
45		eqid 2404	. . . . . . 7 |- ( MetOpen ` ( dist ` U ) ) = ( MetOpen ` ( dist ` U ) )
46	45	mopnval 18421	. . . . . 6 |- ( ( dist ` U ) e. ( * Met ` B ) -> ( MetOpen ` ( dist ` U ) ) = ( topGen ` ran ( ball ` ( dist ` U ) ) ) )
47	15, 46	syl 16	. . . . 5 |- ( ph -> ( MetOpen ` ( dist ` U ) ) = ( topGen ` ran ( ball ` ( dist ` U ) ) ) )
48		xmetf 18312	. . . . . . . 8 |- ( ( dist ` U ) e. ( * Met ` ( Base ` U ) ) -> ( dist ` U ) : ( ( Base ` U ) X. ( Base ` U ) ) --> RR* )
49	20, 48	syl 16	. . . . . . 7 |- ( ph -> ( dist ` U ) : ( ( Base ` U ) X. ( Base ` U ) ) --> RR* )
50		ffn 5550	. . . . . . 7 |- ( ( dist ` U ) : ( ( Base ` U ) X. ( Base ` U ) ) --> RR* -> ( dist ` U ) Fn ( ( Base ` U ) X. ( Base ` U ) ) )
51		fnresdm 5513	. . . . . . 7 |- ( ( dist ` U ) Fn ( ( Base ` U ) X. ( Base ` U ) ) -> ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) = ( dist ` U ) )
52	49, 50, 51	3syl 19	. . . . . 6 |- ( ph -> ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) = ( dist ` U ) )
53	52	fveq2d 5691	. . . . 5 |- ( ph -> ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) = ( MetOpen ` ( dist ` U ) ) )
54	3	ad2antrr 707	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> F : V -1-1-onto-> B )
55		f1of1 5632	. . . . . . . . . . . . . . 15 |- ( F : V -1-1-onto-> B -> F : V -1-1-> B )
56	54, 55	syl 16	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> F : V -1-1-> B )
57		cnvimass 5183	. . . . . . . . . . . . . . 15 |- ( `' F " x ) C_ dom F
58		f1odm 5637	. . . . . . . . . . . . . . . 16 |- ( F : V -1-1-onto-> B -> dom F = V )
59	54, 58	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> dom F = V )
60	57, 59	syl5sseq 3356	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( `' F " x ) C_ V )
61	14	ad2antrr 707	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( ( dist ` R ) |` ( V X. V ) ) e. ( * Met ` V ) )
62		simprl 733	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> y e. V )
63		simprr 734	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> r e. RR* )
64		blssm 18401	. . . . . . . . . . . . . . 15 |- ( ( ( ( dist ` R ) |` ( V X. V ) ) e. ( * Met ` V ) /\ y e. V /\ r e. RR* ) -> ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) C_ V )
65	61, 62, 63, 64	syl3anc 1184	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) C_ V )
66		f1imaeq 5970	. . . . . . . . . . . . . 14 |- ( ( F : V -1-1-> B /\ ( ( `' F " x ) C_ V /\ ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) C_ V ) ) -> ( ( F " ( `' F " x ) ) = ( F " ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) ) <-> ( `' F " x ) = ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) ) )
67	56, 60, 65, 66	syl12anc 1182	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( ( F " ( `' F " x ) ) = ( F " ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) ) <-> ( `' F " x ) = ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) ) )
68	54, 16	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> F : V -onto-> B )
69		simplr 732	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> x C_ B )
70		foimacnv 5651	. . . . . . . . . . . . . . 15 |- ( ( F : V -onto-> B /\ x C_ B ) -> ( F " ( `' F " x ) ) = x )
71	68, 69, 70	syl2anc 643	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( F " ( `' F " x ) ) = x )
72	1	ad2antrr 707	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> U = ( F "s R ) )
73	2	ad2antrr 707	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> V = ( Base ` R ) )
74	4	ad2antrr 707	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> R e. * MetSp )
75	72, 73, 54, 74, 5, 6, 61, 62, 63	imasf1obl 18471	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) = ( F " ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) ) )
76	75	eqcomd 2409	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( F " ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) ) = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) )
77	71, 76	eqeq12d 2418	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( ( F " ( `' F " x ) ) = ( F " ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) ) <-> x = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) ) )
78	67, 77	bitr3d 247	. . . . . . . . . . . 12 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( ( `' F " x ) = ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) <-> x = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) ) )
79	78	2rexbidva 2707	. . . . . . . . . . 11 |- ( ( ph /\ x C_ B ) -> ( E. y e. V E. r e. RR* ( `' F " x ) = ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) <-> E. y e. V E. r e. RR* x = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) ) )
80	3	adantr 452	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ B ) -> F : V -1-1-onto-> B )
81		f1ofn 5634	. . . . . . . . . . . 12 |- ( F : V -1-1-onto-> B -> F Fn V )
82		oveq1 6047	. . . . . . . . . . . . . . 15 |- ( z = ( F ` y ) -> ( z ( ball ` ( dist ` U ) ) r ) = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) )
83	82	eqeq2d 2415	. . . . . . . . . . . . . 14 |- ( z = ( F ` y ) -> ( x = ( z ( ball ` ( dist ` U ) ) r ) <-> x = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) ) )
84	83	rexbidv 2687	. . . . . . . . . . . . 13 |- ( z = ( F ` y ) -> ( E. r e. RR* x = ( z ( ball ` ( dist ` U ) ) r ) <-> E. r e. RR* x = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) ) )
85	84	rexrn 5831	. . . . . . . . . . . 12 |- ( F Fn V -> ( E. z e. ran F E. r e. RR* x = ( z ( ball ` ( dist ` U ) ) r ) <-> E. y e. V E. r e. RR* x = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) ) )
86	80, 81, 85	3syl 19	. . . . . . . . . . 11 |- ( ( ph /\ x C_ B ) -> ( E. z e. ran F E. r e. RR* x = ( z ( ball ` ( dist ` U ) ) r ) <-> E. y e. V E. r e. RR* x = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) ) )
87		forn 5615	. . . . . . . . . . . . 13 |- ( F : V -onto-> B -> ran F = B )
88	80, 16, 87	3syl 19	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ B ) -> ran F = B )
89	88	rexeqdv 2871	. . . . . . . . . . 11 |- ( ( ph /\ x C_ B ) -> ( E. z e. ran F E. r e. RR* x = ( z ( ball ` ( dist ` U ) ) r ) <-> E. z e. B E. r e. RR* x = ( z ( ball ` ( dist ` U ) ) r ) ) )
90	79, 86, 89	3bitr2d 273	. . . . . . . . . 10 |- ( ( ph /\ x C_ B ) -> ( E. y e. V E. r e. RR* ( `' F " x ) = ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) <-> E. z e. B E. r e. RR* x = ( z ( ball ` ( dist ` U ) ) r ) ) )
91	14	adantr 452	. . . . . . . . . . 11 |- ( ( ph /\ x C_ B ) -> ( ( dist ` R ) |` ( V X. V ) ) e. ( * Met ` V ) )
92		blrn 18392	. . . . . . . . . . 11 |- ( ( ( dist ` R ) |` ( V X. V ) ) e. ( * Met ` V ) -> ( ( `' F " x ) e. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) <-> E. y e. V E. r e. RR* ( `' F " x ) = ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) ) )
93	91, 92	syl 16	. . . . . . . . . 10 |- ( ( ph /\ x C_ B ) -> ( ( `' F " x ) e. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) <-> E. y e. V E. r e. RR* ( `' F " x ) = ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) ) )
94	15	adantr 452	. . . . . . . . . . 11 |- ( ( ph /\ x C_ B ) -> ( dist ` U ) e. ( * Met ` B ) )
95		blrn 18392	. . . . . . . . . . 11 |- ( ( dist ` U ) e. ( * Met ` B ) -> ( x e. ran ( ball ` ( dist ` U ) ) <-> E. z e. B E. r e. RR* x = ( z ( ball ` ( dist ` U ) ) r ) ) )
96	94, 95	syl 16	. . . . . . . . . 10 |- ( ( ph /\ x C_ B ) -> ( x e. ran ( ball ` ( dist ` U ) ) <-> E. z e. B E. r e. RR* x = ( z ( ball ` ( dist ` U ) ) r ) ) )
97	90, 93, 96	3bitr4d 277	. . . . . . . . 9 |- ( ( ph /\ x C_ B ) -> ( ( `' F " x ) e. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) <-> x e. ran ( ball ` ( dist ` U ) ) ) )
98	97	pm5.32da 623	. . . . . . . 8 |- ( ph -> ( ( x C_ B /\ ( `' F " x ) e. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) <-> ( x C_ B /\ x e. ran ( ball ` ( dist ` U ) ) ) ) )
99		f1ofo 5640	. . . . . . . . . 10 |- ( F : U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) -1-1-onto-> B -> F : U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) -onto-> B )
100	37, 99	syl 16	. . . . . . . . 9 |- ( ph -> F : U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) -onto-> B )
101	38	elqtop2 17686	. . . . . . . . 9 |- ( ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) e. TopBases /\ F : U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) -onto-> B ) -> ( x e. ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) <-> ( x C_ B /\ ( `' F " x ) e. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) ) )
102	33, 100, 101	syl2anc 643	. . . . . . . 8 |- ( ph -> ( x e. ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) <-> ( x C_ B /\ ( `' F " x ) e. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) ) )
103		blf 18390	. . . . . . . . . . . 12 |- ( ( dist ` U ) e. ( * Met ` B ) -> ( ball ` ( dist ` U ) ) : ( B X. RR* ) --> ~P B )
104		frn 5556	. . . . . . . . . . . 12 |- ( ( ball ` ( dist ` U ) ) : ( B X. RR* ) --> ~P B -> ran ( ball ` ( dist ` U ) ) C_ ~P B )
105	15, 103, 104	3syl 19	. . . . . . . . . . 11 |- ( ph -> ran ( ball ` ( dist ` U ) ) C_ ~P B )
106	105	sseld 3307	. . . . . . . . . 10 |- ( ph -> ( x e. ran ( ball ` ( dist ` U ) ) -> x e. ~P B ) )
107		elpwi 3767	. . . . . . . . . 10 |- ( x e. ~P B -> x C_ B )
108	106, 107	syl6 31	. . . . . . . . 9 |- ( ph -> ( x e. ran ( ball ` ( dist ` U ) ) -> x C_ B ) )
109	108	pm4.71rd 617	. . . . . . . 8 |- ( ph -> ( x e. ran ( ball ` ( dist ` U ) ) <-> ( x C_ B /\ x e. ran ( ball ` ( dist ` U ) ) ) ) )
110	98, 102, 109	3bitr4d 277	. . . . . . 7 |- ( ph -> ( x e. ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) <-> x e. ran ( ball ` ( dist ` U ) ) ) )
111	110	eqrdv 2402	. . . . . 6 |- ( ph -> ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) = ran ( ball ` ( dist ` U ) ) )
112	111	fveq2d 5691	. . . . 5 |- ( ph -> ( topGen ` ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) ) = ( topGen ` ran ( ball ` ( dist ` U ) ) ) )
113	47, 53, 112	3eqtr4d 2446	. . . 4 |- ( ph -> ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) = ( topGen ` ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) ) )
114	40, 44, 113	3eqtr4d 2446	. . 3 |- ( ph -> ( ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) = ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) )
115	26, 31, 114	3eqtrd 2440	. 2 |- ( ph -> ( TopOpen ` U ) = ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) )
116		eqid 2404	. . 3 |- ( Base ` U ) = ( Base ` U )
117		eqid 2404	. . 3 |- ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) = ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) )
118	25, 116, 117	isxms2 18431	. 2 |- ( U e. * MetSp <-> ( ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) e. ( * Met ` ( Base ` U ) ) /\ ( TopOpen ` U ) = ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) ) )
119	23, 115, 118	sylanbrc 646	1 |- ( ph -> U e. * MetSp )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   E.wrex 2667   C_ wss 3280   ~Pcpw 3759   U.cuni 3975   X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838   |` cres 4839   "cima 4840   Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   RR*cxr 9075   Basecbs 13424   distcds 13493   TopOpenctopn 13604   topGenctg 13620   qTop cqtop 13684   "_s cimas 13685   * Metcxmt 16641   ballcbl 16643   MetOpencmopn 16646   TopBasesctb 16917   * MetSpcxme 18300

This theorem is referenced by:  imasf1oms  18473  xpsxms  18517

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-rest 13605  df-topn 13606  df-topgen 13622  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-xms 18303