Theorem imasf1oxms 22294

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 2622	. . . . 5 |- ( ( dist ` R ) |` ( V X. V ) ) = ( ( dist ` R ) |` ( V X. V ) )
6		eqid 2622	. . . . 5 |- ( dist ` U ) = ( dist ` U )
7		eqid 2622	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
8		eqid 2622	. . . . . . . 8 |- ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) )
9	7, 8	xmsxmet 22261	. . . . . . 7 |- ( R e. *MetSp -> ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) e. ( *Met ` ( Base ` R ) ) )
10	4, 9	syl 17	. . . . . 6 |- ( ph -> ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) e. ( *Met ` ( Base ` R ) ) )
11	2	sqxpeqd 5141	. . . . . . 7 |- ( ph -> ( V X. V ) = ( ( Base ` R ) X. ( Base ` R ) ) )
12	11	reseq2d 5396	. . . . . 6 |- ( ph -> ( ( dist ` R ) |` ( V X. V ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) )
13	2	fveq2d 6195	. . . . . 6 |- ( ph -> ( *Met ` V ) = ( *Met ` ( Base ` R ) ) )
14	10, 12, 13	3eltr4d 2716	. . . . 5 |- ( ph -> ( ( dist ` R ) |` ( V X. V ) ) e. ( *Met ` V ) )
15	1, 2, 3, 4, 5, 6, 14	imasf1oxmet 22180	. . . 4 |- ( ph -> ( dist ` U ) e. ( *Met ` B ) )
16		f1ofo 6144	. . . . . . 7 |- ( F : V -1-1-onto-> B -> F : V -onto-> B )
17	3, 16	syl 17	. . . . . 6 |- ( ph -> F : V -onto-> B )
18	1, 2, 17, 4	imasbas 16172	. . . . 5 |- ( ph -> B = ( Base ` U ) )
19	18	fveq2d 6195	. . . 4 |- ( ph -> ( *Met ` B ) = ( *Met ` ( Base ` U ) ) )
20	15, 19	eleqtrd 2703	. . 3 |- ( ph -> ( dist ` U ) e. ( *Met ` ( Base ` U ) ) )
21		ssid 3624	. . 3 |- ( Base ` U ) C_ ( Base ` U )
22		xmetres2 22166	. . 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 694	. 2 |- ( ph -> ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) e. ( *Met ` ( Base ` U ) ) )
24		eqid 2622	. . . 4 |- ( TopOpen ` R ) = ( TopOpen ` R )
25		eqid 2622	. . . 4 |- ( TopOpen ` U ) = ( TopOpen ` U )
26	1, 2, 17, 4, 24, 25	imastopn 21523	. . 3 |- ( ph -> ( TopOpen ` U ) = ( ( TopOpen ` R ) qTop F ) )
27	24, 7, 8	xmstopn 22256	. . . . . 6 |- ( R e. *MetSp -> ( TopOpen ` R ) = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) )
28	4, 27	syl 17	. . . . 5 |- ( ph -> ( TopOpen ` R ) = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) )
29	12	fveq2d 6195	. . . . 5 |- ( ph -> ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) )
30	28, 29	eqtr4d 2659	. . . 4 |- ( ph -> ( TopOpen ` R ) = ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) )
31	30	oveq1d 6665	. . 3 |- ( ph -> ( ( TopOpen ` R ) qTop F ) = ( ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) )
32		blbas 22235	. . . . . 6 |- ( ( ( dist ` R ) |` ( V X. V ) ) e. ( *Met ` V ) -> ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) e. TopBases )
33	14, 32	syl 17	. . . . 5 |- ( ph -> ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) e. TopBases )
34		unirnbl 22225	. . . . . . 7 |- ( ( ( dist ` R ) |` ( V X. V ) ) e. ( *Met ` V ) -> U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) = V )
35		f1oeq2 6128	. . . . . . 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 18	. . . . . 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 247	. . . . 5 |- ( ph -> F : U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) -1-1-onto-> B )
38		eqid 2622	. . . . . 6 |- U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) = U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) )
39	38	tgqtop 21515	. . . . 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 693	. . . 4 |- ( ph -> ( ( topGen ` ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) qTop F ) = ( topGen ` ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) ) )
41		eqid 2622	. . . . . . 7 |- ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) = ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) )
42	41	mopnval 22243	. . . . . 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 17	. . . . 5 |- ( ph -> ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) = ( topGen ` ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) )
44	43	oveq1d 6665	. . . 4 |- ( ph -> ( ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) = ( ( topGen ` ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) qTop F ) )
45		eqid 2622	. . . . . . 7 |- ( MetOpen ` ( dist ` U ) ) = ( MetOpen ` ( dist ` U ) )
46	45	mopnval 22243	. . . . . 6 |- ( ( dist ` U ) e. ( *Met ` B ) -> ( MetOpen ` ( dist ` U ) ) = ( topGen ` ran ( ball ` ( dist ` U ) ) ) )
47	15, 46	syl 17	. . . . 5 |- ( ph -> ( MetOpen ` ( dist ` U ) ) = ( topGen ` ran ( ball ` ( dist ` U ) ) ) )
48		xmetf 22134	. . . . . . . 8 |- ( ( dist ` U ) e. ( *Met ` ( Base ` U ) ) -> ( dist ` U ) : ( ( Base ` U ) X. ( Base ` U ) ) --> RR* )
49	20, 48	syl 17	. . . . . . 7 |- ( ph -> ( dist ` U ) : ( ( Base ` U ) X. ( Base ` U ) ) --> RR* )
50		ffn 6045	. . . . . . 7 |- ( ( dist ` U ) : ( ( Base ` U ) X. ( Base ` U ) ) --> RR* -> ( dist ` U ) Fn ( ( Base ` U ) X. ( Base ` U ) ) )
51		fnresdm 6000	. . . . . . 7 |- ( ( dist ` U ) Fn ( ( Base ` U ) X. ( Base ` U ) ) -> ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) = ( dist ` U ) )
52	49, 50, 51	3syl 18	. . . . . 6 |- ( ph -> ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) = ( dist ` U ) )
53	52	fveq2d 6195	. . . . 5 |- ( ph -> ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) = ( MetOpen ` ( dist ` U ) ) )
54	3	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> F : V -1-1-onto-> B )
55		f1of1 6136	. . . . . . . . . . . . . . 15 |- ( F : V -1-1-onto-> B -> F : V -1-1-> B )
56	54, 55	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> F : V -1-1-> B )
57		cnvimass 5485	. . . . . . . . . . . . . . 15 |- ( `' F " x ) C_ dom F
58		f1odm 6141	. . . . . . . . . . . . . . . 16 |- ( F : V -1-1-onto-> B -> dom F = V )
59	54, 58	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> dom F = V )
60	57, 59	syl5sseq 3653	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( `' F " x ) C_ V )
61	14	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( ( dist ` R ) |` ( V X. V ) ) e. ( *Met ` V ) )
62		simprl 794	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> y e. V )
63		simprr 796	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> r e. RR* )
64		blssm 22223	. . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) C_ V )
66		f1imaeq 6522	. . . . . . . . . . . . . 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 1324	. . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> F : V -onto-> B )
69		simplr 792	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> x C_ B )
70		foimacnv 6154	. . . . . . . . . . . . . . 15 |- ( ( F : V -onto-> B /\ x C_ B ) -> ( F " ( `' F " x ) ) = x )
71	68, 69, 70	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( F " ( `' F " x ) ) = x )
72	1	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> U = ( F "s R ) )
73	2	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> V = ( Base ` R ) )
74	4	ad2antrr 762	. . . . . . . . . . . . . . . 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 22293	. . . . . . . . . . . . . . 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 2628	. . . . . . . . . . . . . 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 2637	. . . . . . . . . . . . 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 270	. . . . . . . . . . . 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 3056	. . . . . . . . . . 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 481	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ B ) -> F : V -1-1-onto-> B )
81		f1ofn 6138	. . . . . . . . . . . 12 |- ( F : V -1-1-onto-> B -> F Fn V )
82		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( z = ( F ` y ) -> ( z ( ball ` ( dist ` U ) ) r ) = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) )
83	82	eqeq2d 2632	. . . . . . . . . . . . . 14 |- ( z = ( F ` y ) -> ( x = ( z ( ball ` ( dist ` U ) ) r ) <-> x = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) ) )
84	83	rexbidv 3052	. . . . . . . . . . . . 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 6361	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 6118	. . . . . . . . . . . . 13 |- ( F : V -onto-> B -> ran F = B )
88	80, 16, 87	3syl 18	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ B ) -> ran F = B )
89	88	rexeqdv 3145	. . . . . . . . . . 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 296	. . . . . . . . . 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 481	. . . . . . . . . . 11 |- ( ( ph /\ x C_ B ) -> ( ( dist ` R ) |` ( V X. V ) ) e. ( *Met ` V ) )
92		blrn 22214	. . . . . . . . . . 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 17	. . . . . . . . . 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 481	. . . . . . . . . . 11 |- ( ( ph /\ x C_ B ) -> ( dist ` U ) e. ( *Met ` B ) )
95		blrn 22214	. . . . . . . . . . 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 17	. . . . . . . . . 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 300	. . . . . . . . 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 673	. . . . . . . 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 6144	. . . . . . . . . 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 17	. . . . . . . . 9 |- ( ph -> F : U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) -onto-> B )
101	38	elqtop2 21504	. . . . . . . . 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 693	. . . . . . . 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 22212	. . . . . . . . . . . 12 |- ( ( dist ` U ) e. ( *Met ` B ) -> ( ball ` ( dist ` U ) ) : ( B X. RR* ) --> ~P B )
104		frn 6053	. . . . . . . . . . . 12 |- ( ( ball ` ( dist ` U ) ) : ( B X. RR* ) --> ~P B -> ran ( ball ` ( dist ` U ) ) C_ ~P B )
105	15, 103, 104	3syl 18	. . . . . . . . . . 11 |- ( ph -> ran ( ball ` ( dist ` U ) ) C_ ~P B )
106	105	sseld 3602	. . . . . . . . . 10 |- ( ph -> ( x e. ran ( ball ` ( dist ` U ) ) -> x e. ~P B ) )
107		elpwi 4168	. . . . . . . . . 10 |- ( x e. ~P B -> x C_ B )
108	106, 107	syl6 35	. . . . . . . . 9 |- ( ph -> ( x e. ran ( ball ` ( dist ` U ) ) -> x C_ B ) )
109	108	pm4.71rd 667	. . . . . . . 8 |- ( ph -> ( x e. ran ( ball ` ( dist ` U ) ) <-> ( x C_ B /\ x e. ran ( ball ` ( dist ` U ) ) ) ) )
110	98, 102, 109	3bitr4d 300	. . . . . . 7 |- ( ph -> ( x e. ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) <-> x e. ran ( ball ` ( dist ` U ) ) ) )
111	110	eqrdv 2620	. . . . . 6 |- ( ph -> ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) = ran ( ball ` ( dist ` U ) ) )
112	111	fveq2d 6195	. . . . 5 |- ( ph -> ( topGen ` ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) ) = ( topGen ` ran ( ball ` ( dist ` U ) ) ) )
113	47, 53, 112	3eqtr4d 2666	. . . 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 2666	. . 3 |- ( ph -> ( ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) = ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) )
115	26, 31, 114	3eqtrd 2660	. 2 |- ( ph -> ( TopOpen ` U ) = ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) )
116		eqid 2622	. . 3 |- ( Base ` U ) = ( Base ` U )
117		eqid 2622	. . 3 |- ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) = ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) )
118	25, 116, 117	isxms2 22253	. 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 698	1 |- ( ph -> U e. *MetSp )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   RR*cxr 10073   Basecbs 15857   distcds 15950   TopOpenctopn 16082   topGenctg 16098   qTop cqtop 16163   "_s cimas 16164   *Metcxmt 19731   ballcbl 19733   MetOpencmopn 19736   TopBasesctb 20749   *MetSpcxme 22122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-xrs 16162  df-qtop 16167  df-imas 16168  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125

This theorem is referenced by:  imasf1oms  22295  xpsxms  22339