Theorem imasf1oxms 20727

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 2467	. . . . 5 |- ( ( dist ` R ) |` ( V X. V ) ) = ( ( dist ` R ) |` ( V X. V ) )
6		eqid 2467	. . . . 5 |- ( dist ` U ) = ( dist ` U )
7		eqid 2467	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
8		eqid 2467	. . . . . . . 8 |- ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) )
9	7, 8	xmsxmet 20694	. . . . . . 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 5024	. . . . . . 7 |- ( ph -> ( V X. V ) = ( ( Base ` R ) X. ( Base ` R ) ) )
12	11	reseq2d 5271	. . . . . 6 |- ( ph -> ( ( dist ` R ) |` ( V X. V ) ) = ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) )
13	2	fveq2d 5868	. . . . . 6 |- ( ph -> ( *Met ` V ) = ( *Met ` ( Base ` R ) ) )
14	10, 12, 13	3eltr4d 2570	. . . . 5 |- ( ph -> ( ( dist ` R ) |` ( V X. V ) ) e. ( *Met ` V ) )
15	1, 2, 3, 4, 5, 6, 14	imasf1oxmet 20613	. . . 4 |- ( ph -> ( dist ` U ) e. ( *Met ` B ) )
16		f1ofo 5821	. . . . . . 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 14763	. . . . 5 |- ( ph -> B = ( Base ` U ) )
19	18	fveq2d 5868	. . . 4 |- ( ph -> ( *Met ` B ) = ( *Met ` ( Base ` U ) ) )
20	15, 19	eleqtrd 2557	. . 3 |- ( ph -> ( dist ` U ) e. ( *Met ` ( Base ` U ) ) )
21		ssid 3523	. . 3 |- ( Base ` U ) C_ ( Base ` U )
22		xmetres2 20599	. . 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 2467	. . . 4 |- ( TopOpen ` R ) = ( TopOpen ` R )
25		eqid 2467	. . . 4 |- ( TopOpen ` U ) = ( TopOpen ` U )
26	1, 2, 17, 4, 24, 25	imastopn 19956	. . 3 |- ( ph -> ( TopOpen ` U ) = ( ( TopOpen ` R ) qTop F ) )
27	24, 7, 8	xmstopn 20689	. . . . . 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 5868	. . . . 5 |- ( ph -> ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) = ( MetOpen ` ( ( dist ` R ) |` ( ( Base ` R ) X. ( Base ` R ) ) ) ) )
30	28, 29	eqtr4d 2511	. . . 4 |- ( ph -> ( TopOpen ` R ) = ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) )
31	30	oveq1d 6297	. . 3 |- ( ph -> ( ( TopOpen ` R ) qTop F ) = ( ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) )
32		blbas 20668	. . . . . 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 20658	. . . . . . 7 |- ( ( ( dist ` R ) |` ( V X. V ) ) e. ( *Met ` V ) -> U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) = V )
35		f1oeq2 5806	. . . . . . 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 2467	. . . . . 6 |- U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) = U. ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) )
39	38	tgqtop 19948	. . . . 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 2467	. . . . . . 7 |- ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) = ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) )
42	41	mopnval 20676	. . . . . 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 6297	. . . 4 |- ( ph -> ( ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) = ( ( topGen ` ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) ) qTop F ) )
45		eqid 2467	. . . . . . 7 |- ( MetOpen ` ( dist ` U ) ) = ( MetOpen ` ( dist ` U ) )
46	45	mopnval 20676	. . . . . 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 20567	. . . . . . . 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 5729	. . . . . . 7 |- ( ( dist ` U ) : ( ( Base ` U ) X. ( Base ` U ) ) --> RR* -> ( dist ` U ) Fn ( ( Base ` U ) X. ( Base ` U ) ) )
51		fnresdm 5688	. . . . . . 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 5868	. . . . 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 5813	. . . . . . . . . . . . . . 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 5355	. . . . . . . . . . . . . . 15 |- ( `' F " x ) C_ dom F
58		f1odm 5818	. . . . . . . . . . . . . . . 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 3552	. . . . . . . . . . . . . 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 20656	. . . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x C_ B ) /\ ( y e. V /\ r e. RR* ) ) -> ( y ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) r ) C_ V )
66		f1imaeq 6159	. . . . . . . . . . . . . 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 1226	. . . . . . . . . . . . 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 5831	. . . . . . . . . . . . . . 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 20726	. . . . . . . . . . . . . . 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 2475	. . . . . . . . . . . . . 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 2489	. . . . . . . . . . . . 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 2979	. . . . . . . . . . 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 5815	. . . . . . . . . . . 12 |- ( F : V -1-1-onto-> B -> F Fn V )
82		oveq1 6289	. . . . . . . . . . . . . . 15 |- ( z = ( F ` y ) -> ( z ( ball ` ( dist ` U ) ) r ) = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) )
83	82	eqeq2d 2481	. . . . . . . . . . . . . 14 |- ( z = ( F ` y ) -> ( x = ( z ( ball ` ( dist ` U ) ) r ) <-> x = ( ( F ` y ) ( ball ` ( dist ` U ) ) r ) ) )
84	83	rexbidv 2973	. . . . . . . . . . . . 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 6021	. . . . . . . . . . . 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 5796	. . . . . . . . . . . . 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 3065	. . . . . . . . . . 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 20647	. . . . . . . . . . 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 20647	. . . . . . . . . . 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 5821	. . . . . . . . . 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 19937	. . . . . . . . 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 20645	. . . . . . . . . . . 12 |- ( ( dist ` U ) e. ( *Met ` B ) -> ( ball ` ( dist ` U ) ) : ( B X. RR* ) --> ~P B )
104		frn 5735	. . . . . . . . . . . 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 3503	. . . . . . . . . 10 |- ( ph -> ( x e. ran ( ball ` ( dist ` U ) ) -> x e. ~P B ) )
107		elpwi 4019	. . . . . . . . . 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 2464	. . . . . 6 |- ( ph -> ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) = ran ( ball ` ( dist ` U ) ) )
112	111	fveq2d 5868	. . . . 5 |- ( ph -> ( topGen ` ( ran ( ball ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) ) = ( topGen ` ran ( ball ` ( dist ` U ) ) ) )
113	47, 53, 112	3eqtr4d 2518	. . . 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 2518	. . 3 |- ( ph -> ( ( MetOpen ` ( ( dist ` R ) |` ( V X. V ) ) ) qTop F ) = ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) )
115	26, 31, 114	3eqtrd 2512	. 2 |- ( ph -> ( TopOpen ` U ) = ( MetOpen ` ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) ) )
116		eqid 2467	. . 3 |- ( Base ` U ) = ( Base ` U )
117		eqid 2467	. . 3 |- ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) ) = ( ( dist ` U ) |` ( ( Base ` U ) X. ( Base ` U ) ) )
118	25, 116, 117	isxms2 20686	. 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 1379   e. wcel 1767   E.wrex 2815   C_ wss 3476   ~Pcpw 4010   U.cuni 4245   X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   |` cres 5001   "cima 5002   Fn wfn 5581   -->wf 5582   -1-1->wf1 5583   -onto->wfo 5584   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   RR*cxr 9623   Basecbs 14486   distcds 14560   TopOpenctopn 14673   topGenctg 14689   qTop cqtop 14754   "_s cimas 14755   *Metcxmt 18174   ballcbl 18176   MetOpencmopn 18179   TopBasesctb 19165   *MetSpcxme 20555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-xrs 14753  df-qtop 14758  df-imas 14759  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-bl 18185  df-mopn 18186  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-xms 20558

This theorem is referenced by:  imasf1oms  20728  xpsxms  20772