Theorem imasf1oxms 20069

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 2443	. . . . 5 |- ( ( dist ` R ) |` ( V X. V ) ) = ( ( dist ` R ) |` ( V X. V ) )
6		eqid 2443	. . . . 5 |- ( dist ` U ) = ( dist ` U )
7		eqid 2443	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
8		eqid 2443	. . . . . . . 8 |- ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) )
9	7, 8	xmsxmet 20036	. . . . . . 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 4870	. . . . . . 7 |- ( ph -> ( V X. V ) = ( ( Base ` R ) X. ( Base ` R ) ) )
12	11	reseq2d 5115	. . . . . 6 |- ( ph -> ( ( dist ` R ) |` ( V X. V ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) )
13	2	fveq2d 5700	. . . . . 6 |- ( ph -> ( *Met ` V ) = ( *Met ` ( Base ` R ) ) )
14	10, 12, 13	3eltr4d 2524	. . . . 5 |- ( ph -> ( ( dist ` R ) |` ( V X. V ) ) e. ( *Met ` V ) )
15	1, 2, 3, 4, 5, 6, 14	imasf1oxmet 19955	. . . 4 |- ( ph -> ( dist ` U ) e. ( *Met ` B ) )
16		f1ofo 5653	. . . . . . 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 14455	. . . . 5 |- ( ph -> B = ( Base ` U ) )
19	18	fveq2d 5700	. . . 4 |- ( ph -> ( *Met ` B ) = ( *Met ` ( Base ` U ) ) )
20	15, 19	eleqtrd 2519	. . 3 |- ( ph -> ( dist ` U ) e. ( *Met ` ( Base ` U ) ) )
21		ssid 3380	. . 3 |- ( Base ` U ) C_ ( Base ` U )
22		xmetres2 19941	. . 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 662	. 2 |- ( ph -> ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) e. ( *Met ` ( Base ` U ) ) )
24		eqid 2443	. . . 4 |- ( TopOpen ` R ) = ( TopOpen ` R )
25		eqid 2443	. . . 4 |- ( TopOpen ` U ) = ( TopOpen ` U )
26	1, 2, 17, 4, 24, 25	imastopn 19298	. . 3 |- ( ph -> ( TopOpen ` U ) = ( ( TopOpen ` R ) qTop F ) )
27	24, 7, 8	xmstopn 20031	. . . . . 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 5700	. . . . 5 |- ( ph -> ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) )
30	28, 29	eqtr4d 2478	. . . 4 |- ( ph -> ( TopOpen ` R ) = ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) )
31	30	oveq1d 6111	. . 3 |- ( ph -> ( ( TopOpen ` R ) qTop F ) = ( ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) )
32		blbas 20010	. . . . . 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 20000	. . . . . . 7 |- ( ( ( dist ` R ) |` ( V X. V ) ) e. ( *Met ` V ) -> U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) = V )
35		f1oeq2 5638	. . . . . . 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 2443	. . . . . 6 |- U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) = U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) )
39	38	tgqtop 19290	. . . . 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 661	. . . 4 |- ( ph -> ( ( topGen ` ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) qTop F ) = ( topGen ` ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) ) )
41		eqid 2443	. . . . . . 7 |- ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) = ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) )
42	41	mopnval 20018	. . . . . 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 6111	. . . 4 |- ( ph -> ( ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) = ( ( topGen ` ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) qTop F ) )
45		eqid 2443	. . . . . . 7 |- ( MetOpen ` ( dist ` U ) ) = ( MetOpen ` ( dist ` U ) )
46	45	mopnval 20018	. . . . . 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 19909	. . . . . . . 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 5564	. . . . . . 7 |- ( ( dist ` U ) : ( ( Base ` U ) X. ( Base ` U ) ) --> RR* -> ( dist ` U ) Fn ( ( Base ` U ) X. ( Base ` U ) ) )
51		fnresdm 5525	. . . . . . 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 5700	. . . . 5 |- ( ph -> ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) = ( MetOpen ` ( dist ` U ) ) )
54	3	ad2antrr 725	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> F : V -1-1-onto-> B )
55		f1of1 5645	. . . . . . . . . . . . . . 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 5194	. . . . . . . . . . . . . . 15 |- ( `' F " x ) C_ dom F
58		f1odm 5650	. . . . . . . . . . . . . . . 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 3409	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( `' F " x ) C_ V )
61	14	ad2antrr 725	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( ( dist ` R ) |` ( V X. V ) ) e. ( *Met ` V ) )
62		simprl 755	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> y e. V )
63		simprr 756	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> r e. RR* )
64		blssm 19998	. . . . . . . . . . . . . . 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 1218	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) C_ V )
66		f1imaeq 5983	. . . . . . . . . . . . . 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 1216	. . . . . . . . . . . . 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 754	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> x C_ B )
70		foimacnv 5663	. . . . . . . . . . . . . . 15 |- ( ( F : V -onto-> B /\ x C_ B ) -> ( F " ( `' F " x ) ) = x )
71	68, 69, 70	syl2anc 661	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( F " ( `' F " x ) ) = x )
72	1	ad2antrr 725	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> U = ( F "s R ) )
73	2	ad2antrr 725	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> V = ( Base ` R ) )
74	4	ad2antrr 725	. . . . . . . . . . . . . . . 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 20068	. . . . . . . . . . . . . . 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 2448	. . . . . . . . . . . . . 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 2457	. . . . . . . . . . . . 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 2761	. . . . . . . . . . 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 465	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ B ) -> F : V -1-1-onto-> B )
81		f1ofn 5647	. . . . . . . . . . . 12 |- ( F : V -1-1-onto-> B -> F Fn V )
82		oveq1 6103	. . . . . . . . . . . . . . 15 |- ( z = ( F ` y ) -> ( z ( ball ` ( dist ` U ) ) r ) = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) )
83	82	eqeq2d 2454	. . . . . . . . . . . . . 14 |- ( z = ( F ` y ) -> ( x = ( z ( ball ` ( dist ` U ) ) r ) <-> x = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) ) )
84	83	rexbidv 2741	. . . . . . . . . . . . 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 5850	. . . . . . . . . . . 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 5628	. . . . . . . . . . . . 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 2929	. . . . . . . . . . 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 465	. . . . . . . . . . 11 |- ( ( ph /\ x C_ B ) -> ( ( dist ` R ) |` ( V X. V ) ) e. ( *Met ` V ) )
92		blrn 19989	. . . . . . . . . . 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 465	. . . . . . . . . . 11 |- ( ( ph /\ x C_ B ) -> ( dist ` U ) e. ( *Met ` B ) )
95		blrn 19989	. . . . . . . . . . 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 641	. . . . . . . 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 5653	. . . . . . . . . 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 19279	. . . . . . . . 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 661	. . . . . . . 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 19987	. . . . . . . . . . . 12 |- ( ( dist ` U ) e. ( *Met ` B ) -> ( ball ` ( dist ` U ) ) : ( B X. RR* ) --> ~P B )
104		frn 5570	. . . . . . . . . . . 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 3360	. . . . . . . . . 10 |- ( ph -> ( x e. ran ( ball ` ( dist ` U ) ) -> x e. ~P B ) )
107		elpwi 3874	. . . . . . . . . 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 635	. . . . . . . 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 2441	. . . . . 6 |- ( ph -> ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) = ran ( ball ` ( dist ` U ) ) )
112	111	fveq2d 5700	. . . . 5 |- ( ph -> ( topGen ` ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) ) = ( topGen ` ran ( ball ` ( dist ` U ) ) ) )
113	47, 53, 112	3eqtr4d 2485	. . . 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 2485	. . 3 |- ( ph -> ( ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) = ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) )
115	26, 31, 114	3eqtrd 2479	. 2 |- ( ph -> ( TopOpen ` U ) = ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) )
116		eqid 2443	. . 3 |- ( Base ` U ) = ( Base ` U )
117		eqid 2443	. . 3 |- ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) = ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) )
118	25, 116, 117	isxms2 20028	. 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 664	1 |- ( ph -> U e. *MetSp )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   E.wrex 2721   C_ wss 3333   ~Pcpw 3865   U.cuni 4096   X. cxp 4843   `'ccnv 4844   dom cdm 4845   ran crn 4846   |` cres 4847   "cima 4848   Fn wfn 5418   -->wf 5419   -1-1->wf1 5420   -onto->wfo 5421   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096   RR*cxr 9422   Basecbs 14179   distcds 14252   TopOpenctopn 14365   topGenctg 14381   qTop cqtop 14446   "_s cimas 14447   *Metcxmt 17806   ballcbl 17808   MetOpencmopn 17811   TopBasesctb 18507   *MetSpcxme 19897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-xrs 14445  df-qtop 14450  df-imas 14451  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-bl 17817  df-mopn 17818  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-xms 19900

This theorem is referenced by:  imasf1oms  20070  xpsxms  20114