Theorem imasf1oxms 20023

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 2441	. . . . 5 |- ( ( dist ` R ) |` ( V X. V ) ) = ( ( dist ` R ) |` ( V X. V ) )
6		eqid 2441	. . . . 5 |- ( dist ` U ) = ( dist ` U )
7		eqid 2441	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
8		eqid 2441	. . . . . . . 8 |- ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) )
9	7, 8	xmsxmet 19990	. . . . . . 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 4861	. . . . . . 7 |- ( ph -> ( V X. V ) = ( ( Base ` R ) X. ( Base ` R ) ) )
12	11	reseq2d 5106	. . . . . 6 |- ( ph -> ( ( dist ` R ) |` ( V X. V ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) )
13	2	fveq2d 5692	. . . . . 6 |- ( ph -> ( *Met ` V ) = ( *Met ` ( Base ` R ) ) )
14	10, 12, 13	3eltr4d 2522	. . . . 5 |- ( ph -> ( ( dist ` R ) |` ( V X. V ) ) e. ( *Met ` V ) )
15	1, 2, 3, 4, 5, 6, 14	imasf1oxmet 19909	. . . 4 |- ( ph -> ( dist ` U ) e. ( *Met ` B ) )
16		f1ofo 5645	. . . . . . 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 14446	. . . . 5 |- ( ph -> B = ( Base ` U ) )
19	18	fveq2d 5692	. . . 4 |- ( ph -> ( *Met ` B ) = ( *Met ` ( Base ` U ) ) )
20	15, 19	eleqtrd 2517	. . 3 |- ( ph -> ( dist ` U ) e. ( *Met ` ( Base ` U ) ) )
21		ssid 3372	. . 3 |- ( Base ` U ) C_ ( Base ` U )
22		xmetres2 19895	. . 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 657	. 2 |- ( ph -> ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) e. ( *Met ` ( Base ` U ) ) )
24		eqid 2441	. . . 4 |- ( TopOpen ` R ) = ( TopOpen ` R )
25		eqid 2441	. . . 4 |- ( TopOpen ` U ) = ( TopOpen ` U )
26	1, 2, 17, 4, 24, 25	imastopn 19252	. . 3 |- ( ph -> ( TopOpen ` U ) = ( ( TopOpen ` R ) qTop F ) )
27	24, 7, 8	xmstopn 19985	. . . . . 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 5692	. . . . 5 |- ( ph -> ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) )
30	28, 29	eqtr4d 2476	. . . 4 |- ( ph -> ( TopOpen ` R ) = ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) )
31	30	oveq1d 6105	. . 3 |- ( ph -> ( ( TopOpen ` R ) qTop F ) = ( ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) )
32		blbas 19964	. . . . . 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 19954	. . . . . . 7 |- ( ( ( dist ` R ) |` ( V X. V ) ) e. ( *Met ` V ) -> U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) = V )
35		f1oeq2 5630	. . . . . . 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 20	. . . . . 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 232	. . . . 5 |- ( ph -> F : U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) -1-1-onto-> B )
38		eqid 2441	. . . . . 6 |- U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) = U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) )
39	38	tgqtop 19244	. . . . 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 656	. . . 4 |- ( ph -> ( ( topGen ` ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) qTop F ) = ( topGen ` ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) ) )
41		eqid 2441	. . . . . . 7 |- ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) = ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) )
42	41	mopnval 19972	. . . . . 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 6105	. . . 4 |- ( ph -> ( ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) = ( ( topGen ` ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) qTop F ) )
45		eqid 2441	. . . . . . 7 |- ( MetOpen ` ( dist ` U ) ) = ( MetOpen ` ( dist ` U ) )
46	45	mopnval 19972	. . . . . 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 19863	. . . . . . . 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 5556	. . . . . . 7 |- ( ( dist ` U ) : ( ( Base ` U ) X. ( Base ` U ) ) --> RR* -> ( dist ` U ) Fn ( ( Base ` U ) X. ( Base ` U ) ) )
51		fnresdm 5517	. . . . . . 7 |- ( ( dist ` U ) Fn ( ( Base ` U ) X. ( Base ` U ) ) -> ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) = ( dist ` U ) )
52	49, 50, 51	3syl 20	. . . . . 6 |- ( ph -> ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) = ( dist ` U ) )
53	52	fveq2d 5692	. . . . 5 |- ( ph -> ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) = ( MetOpen ` ( dist ` U ) ) )
54	3	ad2antrr 720	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> F : V -1-1-onto-> B )
55		f1of1 5637	. . . . . . . . . . . . . . 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 5186	. . . . . . . . . . . . . . 15 |- ( `' F " x ) C_ dom F
58		f1odm 5642	. . . . . . . . . . . . . . . 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 3401	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( `' F " x ) C_ V )
61	14	ad2antrr 720	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( ( dist ` R ) |` ( V X. V ) ) e. ( *Met ` V ) )
62		simprl 750	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> y e. V )
63		simprr 751	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> r e. RR* )
64		blssm 19952	. . . . . . . . . . . . . . 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 1213	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) C_ V )
66		f1imaeq 5975	. . . . . . . . . . . . . 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 1211	. . . . . . . . . . . . 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 749	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> x C_ B )
70		foimacnv 5655	. . . . . . . . . . . . . . 15 |- ( ( F : V -onto-> B /\ x C_ B ) -> ( F " ( `' F " x ) ) = x )
71	68, 69, 70	syl2anc 656	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( F " ( `' F " x ) ) = x )
72	1	ad2antrr 720	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> U = ( F "s R ) )
73	2	ad2antrr 720	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> V = ( Base ` R ) )
74	4	ad2antrr 720	. . . . . . . . . . . . . . . 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 20022	. . . . . . . . . . . . . . 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 2446	. . . . . . . . . . . . . 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 2455	. . . . . . . . . . . . 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 255	. . . . . . . . . . . 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 2754	. . . . . . . . . . 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 462	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ B ) -> F : V -1-1-onto-> B )
81		f1ofn 5639	. . . . . . . . . . . 12 |- ( F : V -1-1-onto-> B -> F Fn V )
82		oveq1 6097	. . . . . . . . . . . . . . 15 |- ( z = ( F ` y ) -> ( z ( ball ` ( dist ` U ) ) r ) = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) )
83	82	eqeq2d 2452	. . . . . . . . . . . . . 14 |- ( z = ( F ` y ) -> ( x = ( z ( ball ` ( dist ` U ) ) r ) <-> x = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) ) )
84	83	rexbidv 2734	. . . . . . . . . . . . 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 5842	. . . . . . . . . . . 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 20	. . . . . . . . . . 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 5620	. . . . . . . . . . . . 13 |- ( F : V -onto-> B -> ran F = B )
88	80, 16, 87	3syl 20	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ B ) -> ran F = B )
89	88	rexeqdv 2922	. . . . . . . . . . 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 281	. . . . . . . . . 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 462	. . . . . . . . . . 11 |- ( ( ph /\ x C_ B ) -> ( ( dist ` R ) |` ( V X. V ) ) e. ( *Met ` V ) )
92		blrn 19943	. . . . . . . . . . 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 462	. . . . . . . . . . 11 |- ( ( ph /\ x C_ B ) -> ( dist ` U ) e. ( *Met ` B ) )
95		blrn 19943	. . . . . . . . . . 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 285	. . . . . . . . 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 636	. . . . . . . 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 5645	. . . . . . . . . 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 19233	. . . . . . . . 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 656	. . . . . . . 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 19941	. . . . . . . . . . . 12 |- ( ( dist ` U ) e. ( *Met ` B ) -> ( ball ` ( dist ` U ) ) : ( B X. RR* ) --> ~P B )
104		frn 5562	. . . . . . . . . . . 12 |- ( ( ball ` ( dist ` U ) ) : ( B X. RR* ) --> ~P B -> ran ( ball ` ( dist ` U ) ) C_ ~P B )
105	15, 103, 104	3syl 20	. . . . . . . . . . 11 |- ( ph -> ran ( ball ` ( dist ` U ) ) C_ ~P B )
106	105	sseld 3352	. . . . . . . . . 10 |- ( ph -> ( x e. ran ( ball ` ( dist ` U ) ) -> x e. ~P B ) )
107		elpwi 3866	. . . . . . . . . 10 |- ( x e. ~P B -> x C_ B )
108	106, 107	syl6 33	. . . . . . . . 9 |- ( ph -> ( x e. ran ( ball ` ( dist ` U ) ) -> x C_ B ) )
109	108	pm4.71rd 630	. . . . . . . 8 |- ( ph -> ( x e. ran ( ball ` ( dist ` U ) ) <-> ( x C_ B /\ x e. ran ( ball ` ( dist ` U ) ) ) ) )
110	98, 102, 109	3bitr4d 285	. . . . . . 7 |- ( ph -> ( x e. ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) <-> x e. ran ( ball ` ( dist ` U ) ) ) )
111	110	eqrdv 2439	. . . . . 6 |- ( ph -> ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) = ran ( ball ` ( dist ` U ) ) )
112	111	fveq2d 5692	. . . . 5 |- ( ph -> ( topGen ` ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) ) = ( topGen ` ran ( ball ` ( dist ` U ) ) ) )
113	47, 53, 112	3eqtr4d 2483	. . . 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 2483	. . 3 |- ( ph -> ( ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) = ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) )
115	26, 31, 114	3eqtrd 2477	. 2 |- ( ph -> ( TopOpen ` U ) = ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) )
116		eqid 2441	. . 3 |- ( Base ` U ) = ( Base ` U )
117		eqid 2441	. . 3 |- ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) = ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) )
118	25, 116, 117	isxms2 19982	. 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 659	1 |- ( ph -> U e. *MetSp )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1364   e. wcel 1761   E.wrex 2714   C_ wss 3325   ~Pcpw 3857   U.cuni 4088   X. cxp 4834   `'ccnv 4835   dom cdm 4836   ran crn 4837   |` cres 4838   "cima 4839   Fn wfn 5410   -->wf 5411   -1-1->wf1 5412   -onto->wfo 5413   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   RR*cxr 9413   Basecbs 14170   distcds 14243   TopOpenctopn 14356   topGenctg 14372   qTop cqtop 14437   "_s cimas 14438   *Metcxmt 17760   ballcbl 17762   MetOpencmopn 17765   TopBasesctb 18461   *MetSpcxme 19851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-xrs 14436  df-qtop 14441  df-imas 14442  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-bl 17771  df-mopn 17772  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-xms 19854

This theorem is referenced by:  imasf1oms  20024  xpsxms  20068