Theorem metcnp 9165

Description: Two ways to say a mapping from metric C to metric D is continuous at point P. Warning: The HTML proof page is 0.6 megabyte in size.

Hypotheses

Ref	Expression
metcn.1	|- X = dom dom C
metcn.2	|- J = (Open` C)
metcn.3	|- Y = dom dom D
metcn.4	|- K = (Open` D)

Assertion

Ref	Expression
metcnp	|- ((C e. Met /\ D e. Met /\ P e. X) -> (F e. ((J CnP K)` P) <-> (F:X-->Y /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. X ((PCw) < z -> ((F` P)D(F` w)) < y))))))

Distinct variable groups:   y,w,z,F   w,C,y,z   w,D,y,z   w,X,y,z   w,Y,y,z   w,P,y,z   w,J,y,z   y,K

Proof of Theorem metcnp

Step	Hyp	Ref	Expression
1		metcn.2	. . . . 5 |- J = (Open` C)
2	1	opntop 9147	. . . 4 |- (C e. Met -> J e. Top)
3	2	3ad2ant1 897	. . 3 |- ((C e. Met /\ D e. Met /\ P e. X) -> J e. Top)
4		metcn.4	. . . . 5 |- K = (Open` D)
5	4	opntop 9147	. . . 4 |- (D e. Met -> K e. Top)
6	5	3ad2ant2 898	. . 3 |- ((C e. Met /\ D e. Met /\ P e. X) -> K e. Top)
7		metcn.1	. . . . . . 7 |- X = dom dom C
8	7, 1	uniopn2 9138	. . . . . 6 |- (C e. Met -> U.J = X)
9	8	eleq2d 1964	. . . . 5 |- (C e. Met -> (P e. U.J <-> P e. X))
10	9	biimpar 461	. . . 4 |- ((C e. Met /\ P e. X) -> P e. U.J)
11	10	3adant2 895	. . 3 |- ((C e. Met /\ D e. Met /\ P e. X) -> P e. U.J)
12		eqid 1884	. . . 4 |- U.J = U.J
13		eqid 1884	. . . 4 |- U.K = U.K
14	12, 13	iscnp 9036	. . 3 |- ((J e. Top /\ K e. Top /\ P e. U.J) -> (F e. ((J CnP K)` P) <-> (F:U.J-->U.K /\ A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)))))
15	3, 6, 11, 14	syl111anc 1100	. 2 |- ((C e. Met /\ D e. Met /\ P e. X) -> (F e. ((J CnP K)` P) <-> (F:U.J-->U.K /\ A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)))))
16		feq23 4554	. . . . 5 |- ((U.J = X /\ U.K = Y) -> (F:U.J-->U.K <-> F:X-->Y))
17		metcn.3	. . . . . 6 |- Y = dom dom D
18	17, 4	uniopn2 9138	. . . . 5 |- (D e. Met -> U.K = Y)
19	16, 8, 18	syl2an 503	. . . 4 |- ((C e. Met /\ D e. Met) -> (F:U.J-->U.K <-> F:X-->Y))
20	19	anbi1d 679	. . 3 |- ((C e. Met /\ D e. Met) -> ((F:U.J-->U.K /\ A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v))) <-> (F:X-->Y /\ A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)))))
21	20	3adant3 896	. 2 |- ((C e. Met /\ D e. Met /\ P e. X) -> ((F:U.J-->U.K /\ A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v))) <-> (F:X-->Y /\ A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)))))
22	17, 4	blopn 9153	. . . . . . . . . . . . . . . 16 |- (((D e. Met /\ (F` P) e. Y) /\ (y e. RR /\ 0 < y)) -> ((F` P)( ball ` D)y) e. K)
23	17	blcntr 9122	. . . . . . . . . . . . . . . 16 |- (((D e. Met /\ (F` P) e. Y) /\ (y e. RR /\ 0 < y)) -> (F` P) e. ((F` P)( ball ` D)y))
24		eleq2 1958	. . . . . . . . . . . . . . . . . . 19 |- (v = ((F` P)( ball ` D)y) -> ((F` P) e. v <-> (F` P) e. ((F` P)( ball ` D)y)))
25		sseq2 2639	. . . . . . . . . . . . . . . . . . . . 21 |- (v = ((F` P)( ball ` D)y) -> ((F"u) C_ v <-> (F"u) C_ ((F` P)( ball ` D)y)))
26	25	anbi2d 678	. . . . . . . . . . . . . . . . . . . 20 |- (v = ((F` P)( ball ` D)y) -> ((P e. u /\ (F"u) C_ v) <-> (P e. u /\ (F"u) C_ ((F` P)( ball ` D)y))))
27	26	rexbidv 2124	. . . . . . . . . . . . . . . . . . 19 |- (v = ((F` P)( ball ` D)y) -> (E.u e. J (P e. u /\ (F"u) C_ v) <-> E.u e. J (P e. u /\ (F"u) C_ ((F` P)( ball ` D)y))))
28	24, 27	imbi12d 688	. . . . . . . . . . . . . . . . . 18 |- (v = ((F` P)( ball ` D)y) -> (((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)) <-> ((F` P) e. ((F` P)( ball ` D)y) -> E.u e. J (P e. u /\ (F"u) C_ ((F` P)( ball ` D)y)))))
29	28	rcla4v 2376	. . . . . . . . . . . . . . . . 17 |- (((F` P)( ball ` D)y) e. K -> (A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)) -> ((F` P) e. ((F` P)( ball ` D)y) -> E.u e. J (P e. u /\ (F"u) C_ ((F` P)( ball ` D)y)))))
30	29	com23 36	. . . . . . . . . . . . . . . 16 |- (((F` P)( ball ` D)y) e. K -> ((F` P) e. ((F` P)( ball ` D)y) -> (A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)) -> E.u e. J (P e. u /\ (F"u) C_ ((F` P)( ball ` D)y)))))
31	22, 23, 30	sylc 83	. . . . . . . . . . . . . . 15 |- (((D e. Met /\ (F` P) e. Y) /\ (y e. RR /\ 0 < y)) -> (A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)) -> E.u e. J (P e. u /\ (F"u) C_ ((F` P)( ball ` D)y))))
32		ffvelrn 4787	. . . . . . . . . . . . . . 15 |- ((F:X-->Y /\ P e. X) -> (F` P) e. Y)
33	31, 32	sylanl2 510	. . . . . . . . . . . . . 14 |- (((D e. Met /\ (F:X-->Y /\ P e. X)) /\ (y e. RR /\ 0 < y)) -> (A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)) -> E.u e. J (P e. u /\ (F"u) C_ ((F` P)( ball ` D)y))))
34	33	adantlll 432	. . . . . . . . . . . . 13 |- ((((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) /\ (y e. RR /\ 0 < y)) -> (A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)) -> E.u e. J (P e. u /\ (F"u) C_ ((F` P)( ball ` D)y))))
35		ffun 4565	. . . . . . . . . . . . . . . . . . . . 21 |- (F:X-->Y -> Fun F)
36	35	ad2antlr 441	. . . . . . . . . . . . . . . . . . . 20 |- (((C e. Met /\ F:X-->Y) /\ u e. J) -> Fun F)
37	7, 1	opnss 9140	. . . . . . . . . . . . . . . . . . . . . 22 |- ((C e. Met /\ u e. J) -> u C_ X)
38	37	adantlr 429	. . . . . . . . . . . . . . . . . . . . 21 |- (((C e. Met /\ F:X-->Y) /\ u e. J) -> u C_ X)
39		fdm 4567	. . . . . . . . . . . . . . . . . . . . . 22 |- (F:X-->Y -> dom F = X)
40	39	ad2antlr 441	. . . . . . . . . . . . . . . . . . . . 21 |- (((C e. Met /\ F:X-->Y) /\ u e. J) -> dom F = X)
41	38, 40	sseqtr4d 2654	. . . . . . . . . . . . . . . . . . . 20 |- (((C e. Met /\ F:X-->Y) /\ u e. J) -> u C_ dom F)
42		funimass3 4779	. . . . . . . . . . . . . . . . . . . 20 |- ((Fun F /\ u C_ dom F) -> ((F"u) C_ ((F` P)( ball ` D)y) <-> u C_ (`'F"((F` P)( ball ` D)y))))
43	36, 41, 42	syl11anc 524	. . . . . . . . . . . . . . . . . . 19 |- (((C e. Met /\ F:X-->Y) /\ u e. J) -> ((F"u) C_ ((F` P)( ball ` D)y) <-> u C_ (`'F"((F` P)( ball ` D)y))))
44	43	adantr 425	. . . . . . . . . . . . . . . . . 18 |- ((((C e. Met /\ F:X-->Y) /\ u e. J) /\ P e. u) -> ((F"u) C_ ((F` P)( ball ` D)y) <-> u C_ (`'F"((F` P)( ball ` D)y))))
45		sstr 2625	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((P( ball ` C)z) C_ u /\ u C_ (`'F"((F` P)( ball ` D)y))) -> (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))
46	45	expcom 403	. . . . . . . . . . . . . . . . . . . . . . 23 |- (u C_ (`'F"((F` P)( ball ` D)y)) -> ((P( ball ` C)z) C_ u -> (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))
47	46	adantr 425	. . . . . . . . . . . . . . . . . . . . . 22 |- ((u C_ (`'F"((F` P)( ball ` D)y)) /\ z e. RR) -> ((P( ball ` C)z) C_ u -> (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))
48	47	anim2d 620	. . . . . . . . . . . . . . . . . . . . 21 |- ((u C_ (`'F"((F` P)( ball ` D)y)) /\ z e. RR) -> ((0 < z /\ (P( ball ` C)z) C_ u) -> (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))))
49	48	reximdva 2203	. . . . . . . . . . . . . . . . . . . 20 |- (u C_ (`'F"((F` P)( ball ` D)y)) -> (E.z e. RR (0 < z /\ (P( ball ` C)z) C_ u) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))))
50	1	opni2 9142	. . . . . . . . . . . . . . . . . . . . 21 |- ((C e. Met /\ u e. J /\ P e. u) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ u))
51	50	3expa 1067	. . . . . . . . . . . . . . . . . . . 20 |- (((C e. Met /\ u e. J) /\ P e. u) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ u))
52	49, 51	syl5com 63	. . . . . . . . . . . . . . . . . . 19 |- (((C e. Met /\ u e. J) /\ P e. u) -> (u C_ (`'F"((F` P)( ball ` D)y)) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))))
53	52	adantllr 433	. . . . . . . . . . . . . . . . . 18 |- ((((C e. Met /\ F:X-->Y) /\ u e. J) /\ P e. u) -> (u C_ (`'F"((F` P)( ball ` D)y)) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))))
54	44, 53	sylbid 220	. . . . . . . . . . . . . . . . 17 |- ((((C e. Met /\ F:X-->Y) /\ u e. J) /\ P e. u) -> ((F"u) C_ ((F` P)( ball ` D)y) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))))
55	54	expimpd 404	. . . . . . . . . . . . . . . 16 |- (((C e. Met /\ F:X-->Y) /\ u e. J) -> ((P e. u /\ (F"u) C_ ((F` P)( ball ` D)y)) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))))
56	55	r19.23adva 2216	. . . . . . . . . . . . . . 15 |- ((C e. Met /\ F:X-->Y) -> (E.u e. J (P e. u /\ (F"u) C_ ((F` P)( ball ` D)y)) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))))
57	56	ad2ant2r 445	. . . . . . . . . . . . . 14 |- (((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) -> (E.u e. J (P e. u /\ (F"u) C_ ((F` P)( ball ` D)y)) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))))
58	57	adantr 425	. . . . . . . . . . . . 13 |- ((((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) /\ (y e. RR /\ 0 < y)) -> (E.u e. J (P e. u /\ (F"u) C_ ((F` P)( ball ` D)y)) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))))
59	34, 58	syld 30	. . . . . . . . . . . 12 |- ((((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) /\ (y e. RR /\ 0 < y)) -> (A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))))
60	59	imp 377	. . . . . . . . . . 11 |- (((((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) /\ (y e. RR /\ 0 < y)) /\ A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v))) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))
61	60	an1rs 547	. . . . . . . . . 10 |- (((((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) /\ A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v))) /\ (y e. RR /\ 0 < y)) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))
62	61	exp32 408	. . . . . . . . 9 |- ((((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) /\ A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v))) -> (y e. RR -> (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))))
63	62	r19.21aiv 2175	. . . . . . . 8 |- ((((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) /\ A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v))) -> A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))))
64	4	opni2 9142	. . . . . . . . . . . . . 14 |- ((D e. Met /\ v e. K /\ (F` P) e. v) -> E.w e. RR (0 < w /\ ((F` P)( ball ` D)w) C_ v))
65	64	3expb 1068	. . . . . . . . . . . . 13 |- ((D e. Met /\ (v e. K /\ (F` P) e. v)) -> E.w e. RR (0 < w /\ ((F` P)( ball ` D)w) C_ v))
66		simpllr 453	. . . . . . . . . . . . 13 |- ((((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))) -> D e. Met)
67	65, 66	sylan 497	. . . . . . . . . . . 12 |- (((((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))) /\ (v e. K /\ (F` P) e. v)) -> E.w e. RR (0 < w /\ ((F` P)( ball ` D)w) C_ v))
68	67	anassrs 489	. . . . . . . . . . 11 |- ((((((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))) /\ v e. K) /\ (F` P) e. v) -> E.w e. RR (0 < w /\ ((F` P)( ball ` D)w) C_ v))
69		breq2 3342	. . . . . . . . . . . . . . . . . . . . 21 |- (y = w -> (0 < y <-> 0 < w))
70		opreq2 4890	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (y = w -> ((F` P)( ball ` D)y) = ((F` P)( ball ` D)w))
71	70	imaeq2d 4264	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y = w -> (`'F"((F` P)( ball ` D)y)) = (`'F"((F` P)( ball ` D)w)))
72	71	sseq2d 2645	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y = w -> ((P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)) <-> (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w))))
73	72	anbi2d 678	. . . . . . . . . . . . . . . . . . . . . 22 |- (y = w -> ((0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))) <-> (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w)))))
74	73	rexbidv 2124	. . . . . . . . . . . . . . . . . . . . 21 |- (y = w -> (E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))) <-> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w)))))
75	69, 74	imbi12d 688	. . . . . . . . . . . . . . . . . . . 20 |- (y = w -> ((0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))) <-> (0 < w -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w))))))
76	75	rcla4cva 2379	. . . . . . . . . . . . . . . . . . 19 |- ((A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))) /\ w e. RR) -> (0 < w -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w)))))
77	76	imp 377	. . . . . . . . . . . . . . . . . 18 |- (((A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))) /\ w e. RR) /\ 0 < w) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w))))
78	77	adantlll 432	. . . . . . . . . . . . . . . . 17 |- (((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))) /\ w e. RR) /\ 0 < w) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w))))
79	78	adantrr 431	. . . . . . . . . . . . . . . 16 |- (((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))) /\ w e. RR) /\ (0 < w /\ ((F` P)( ball ` D)w) C_ v)) -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w))))
80	35	ad2antrl 442	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((C e. Met /\ (F:X-->Y /\ P e. X)) -> Fun F)
81	80	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ z e. RR) /\ 0 < z) -> Fun F)
82	7	blssm 9127	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (((C e. Met /\ P e. X) /\ (z e. RR /\ 0 < z)) -> (P( ball ` C)z) C_ X)
83	82	adantlrl 434	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (((C e. Met /\ (F:X-->Y /\ P e. X)) /\ (z e. RR /\ 0 < z)) -> (P( ball ` C)z) C_ X)
84	83	anassrs 489	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ z e. RR) /\ 0 < z) -> (P( ball ` C)z) C_ X)
85	39	ad2antrl 442	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((C e. Met /\ (F:X-->Y /\ P e. X)) -> dom F = X)
86	85	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ z e. RR) /\ 0 < z) -> dom F = X)
87	84, 86	sseqtr4d 2654	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ z e. RR) /\ 0 < z) -> (P( ball ` C)z) C_ dom F)
88		funimass3 4779	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((Fun F /\ (P( ball ` C)z) C_ dom F) -> ((F"(P( ball ` C)z)) C_ ((F` P)( ball ` D)w) <-> (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w))))
89	81, 87, 88	syl11anc 524	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ z e. RR) /\ 0 < z) -> ((F"(P( ball ` C)z)) C_ ((F` P)( ball ` D)w) <-> (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w))))
90	89	adantllr 433	. . . . . . . . . . . . . . . . . . . . . 22 |- (((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ ((F` P)( ball ` D)w) C_ v) /\ z e. RR) /\ 0 < z) -> ((F"(P( ball ` C)z)) C_ ((F` P)( ball ` D)w) <-> (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w))))
91	7, 1	blopn 9153	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (((C e. Met /\ P e. X) /\ (z e. RR /\ 0 < z)) -> (P( ball ` C)z) e. J)
92	91	adantlrl 434	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (((C e. Met /\ (F:X-->Y /\ P e. X)) /\ (z e. RR /\ 0 < z)) -> (P( ball ` C)z) e. J)
93	92	anassrs 489	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ z e. RR) /\ 0 < z) -> (P( ball ` C)z) e. J)
94	93	adantllr 433	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ ((F` P)( ball ` D)w) C_ v) /\ z e. RR) /\ 0 < z) -> (P( ball ` C)z) e. J)
95	94	adantr 425	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ ((F` P)( ball ` D)w) C_ v) /\ z e. RR) /\ 0 < z) /\ (F"(P( ball ` C)z)) C_ ((F` P)( ball ` D)w)) -> (P( ball ` C)z) e. J)
96	7	blcntr 9122	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (((C e. Met /\ P e. X) /\ (z e. RR /\ 0 < z)) -> P e. (P( ball ` C)z))
97	96	adantlrl 434	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (((C e. Met /\ (F:X-->Y /\ P e. X)) /\ (z e. RR /\ 0 < z)) -> P e. (P( ball ` C)z))
98	97	anassrs 489	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ z e. RR) /\ 0 < z) -> P e. (P( ball ` C)z))
99	98	adantllr 433	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ ((F` P)( ball ` D)w) C_ v) /\ z e. RR) /\ 0 < z) -> P e. (P( ball ` C)z))
100	99	adantr 425	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ ((F` P)( ball ` D)w) C_ v) /\ z e. RR) /\ 0 < z) /\ (F"(P( ball ` C)z)) C_ ((F` P)( ball ` D)w)) -> P e. (P( ball ` C)z))
101		sstr 2625	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (((F"(P( ball ` C)z)) C_ ((F` P)( ball ` D)w) /\ ((F` P)( ball ` D)w) C_ v) -> (F"(P( ball ` C)z)) C_ v)
102	101	ancoms 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((((F` P)( ball ` D)w) C_ v /\ (F"(P( ball ` C)z)) C_ ((F` P)( ball ` D)w)) -> (F"(P( ball ` C)z)) C_ v)
103	102	adantlr 429	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (((((F` P)( ball ` D)w) C_ v /\ 0 < z) /\ (F"(P( ball ` C)z)) C_ ((F` P)( ball ` D)w)) -> (F"(P( ball ` C)z)) C_ v)
104	103	adantlll 432	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ ((F` P)( ball ` D)w) C_ v) /\ 0 < z) /\ (F"(P( ball ` C)z)) C_ ((F` P)( ball ` D)w)) -> (F"(P( ball ` C)z)) C_ v)
105	104	adantllr 433	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ ((F` P)( ball ` D)w) C_ v) /\ z e. RR) /\ 0 < z) /\ (F"(P( ball ` C)z)) C_ ((F` P)( ball ` D)w)) -> (F"(P( ball ` C)z)) C_ v)
106		eleq2 1958	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (u = (P( ball ` C)z) -> (P e. u <-> P e. (P( ball ` C)z)))
107		imaeq2 4260	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (u = (P( ball ` C)z) -> (F"u) = (F"(P( ball ` C)z)))
108	107	sseq1d 2644	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (u = (P( ball ` C)z) -> ((F"u) C_ v <-> (F"(P( ball ` C)z)) C_ v))
109	106, 108	anbi12d 690	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (u = (P( ball ` C)z) -> ((P e. u /\ (F"u) C_ v) <-> (P e. (P( ball ` C)z) /\ (F"(P( ball ` C)z)) C_ v)))
110	109	rcla4ev 2381	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((P( ball ` C)z) e. J /\ (P e. (P( ball ` C)z) /\ (F"(P( ball ` C)z)) C_ v)) -> E.u e. J (P e. u /\ (F"u) C_ v))
111	95, 100, 105, 110	syl12anc 1098	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ ((F` P)( ball ` D)w) C_ v) /\ z e. RR) /\ 0 < z) /\ (F"(P( ball ` C)z)) C_ ((F` P)( ball ` D)w)) -> E.u e. J (P e. u /\ (F"u) C_ v))
112	111	ex 402	. . . . . . . . . . . . . . . . . . . . . 22 |- (((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ ((F` P)( ball ` D)w) C_ v) /\ z e. RR) /\ 0 < z) -> ((F"(P( ball ` C)z)) C_ ((F` P)( ball ` D)w) -> E.u e. J (P e. u /\ (F"u) C_ v)))
113	90, 112	sylbird 222	. . . . . . . . . . . . . . . . . . . . 21 |- (((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ ((F` P)( ball ` D)w) C_ v) /\ z e. RR) /\ 0 < z) -> ((P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w)) -> E.u e. J (P e. u /\ (F"u) C_ v)))
114	113	expimpd 404	. . . . . . . . . . . . . . . . . . . 20 |- ((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ ((F` P)( ball ` D)w) C_ v) /\ z e. RR) -> ((0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w))) -> E.u e. J (P e. u /\ (F"u) C_ v)))
115	114	r19.23adva 2216	. . . . . . . . . . . . . . . . . . 19 |- (((C e. Met /\ (F:X-->Y /\ P e. X)) /\ ((F` P)( ball ` D)w) C_ v) -> (E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w))) -> E.u e. J (P e. u /\ (F"u) C_ v)))
116	115	adantlr 429	. . . . . . . . . . . . . . . . . 18 |- ((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ w e. RR) /\ ((F` P)( ball ` D)w) C_ v) -> (E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w))) -> E.u e. J (P e. u /\ (F"u) C_ v)))
117	116	adantllr 433	. . . . . . . . . . . . . . . . 17 |- (((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))) /\ w e. RR) /\ ((F` P)( ball ` D)w) C_ v) -> (E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w))) -> E.u e. J (P e. u /\ (F"u) C_ v)))
118	117	adantrl 430	. . . . . . . . . . . . . . . 16 |- (((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))) /\ w e. RR) /\ (0 < w /\ ((F` P)( ball ` D)w) C_ v)) -> (E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)w))) -> E.u e. J (P e. u /\ (F"u) C_ v)))
119	79, 118	mpd 29	. . . . . . . . . . . . . . 15 |- (((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))) /\ w e. RR) /\ (0 < w /\ ((F` P)( ball ` D)w) C_ v)) -> E.u e. J (P e. u /\ (F"u) C_ v))
120	119	ex 402	. . . . . . . . . . . . . 14 |- ((((C e. Met /\ (F:X-->Y /\ P e. X)) /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))) /\ w e. RR) -> ((0 < w /\ ((F` P)( ball ` D)w) C_ v) -> E.u e. J (P e. u /\ (F"u) C_ v)))
121	120	r19.23adva 2216	. . . . . . . . . . . . 13 |- (((C e. Met /\ (F:X-->Y /\ P e. X)) /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))) -> (E.w e. RR (0 < w /\ ((F` P)( ball ` D)w) C_ v) -> E.u e. J (P e. u /\ (F"u) C_ v)))
122	121	adantllr 433	. . . . . . . . . . . 12 |- ((((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))) -> (E.w e. RR (0 < w /\ ((F` P)( ball ` D)w) C_ v) -> E.u e. J (P e. u /\ (F"u) C_ v)))
123	122	ad2antrr 440	. . . . . . . . . . 11 |- ((((((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))) /\ v e. K) /\ (F` P) e. v) -> (E.w e. RR (0 < w /\ ((F` P)( ball ` D)w) C_ v) -> E.u e. J (P e. u /\ (F"u) C_ v)))
124	68, 123	mpd 29	. . . . . . . . . 10 |- ((((((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))) /\ v e. K) /\ (F` P) e. v) -> E.u e. J (P e. u /\ (F"u) C_ v))
125	124	ex 402	. . . . . . . . 9 |- (((((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))) /\ v e. K) -> ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)))
126	125	r19.21aiva 2176	. . . . . . . 8 |- ((((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))) -> A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)))
127	63, 126	impbida 577	. . . . . . 7 |- (((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) -> (A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)) <-> A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y))))))
128	7, 1, 17, 4	metcnplem 9164	. . . . . . 7 |- (((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) -> (A.y e. RR (0 < y -> E.z e. RR (0 < z /\ (P( ball ` C)z) C_ (`'F"((F` P)( ball ` D)y)))) <-> A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. X ((PCw) < z -> ((F` P)D(F` w)) < y)))))
129	127, 128	bitrd 587	. . . . . 6 |- (((C e. Met /\ D e. Met) /\ (F:X-->Y /\ P e. X)) -> (A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)) <-> A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. X ((PCw) < z -> ((F` P)D(F` w)) < y)))))
130	129	exp43 415	. . . . 5 |- (C e. Met -> (D e. Met -> (F:X-->Y -> (P e. X -> (A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)) <-> A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. X ((PCw) < z -> ((F` P)D(F` w)) < y))))))))
131	130	com34 40	. . . 4 |- (C e. Met -> (D e. Met -> (P e. X -> (F:X-->Y -> (A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)) <-> A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. X ((PCw) < z -> ((F` P)D(F` w)) < y))))))))
132	131	3imp 1061	. . 3 |- ((C e. Met /\ D e. Met /\ P e. X) -> (F:X-->Y -> (A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v)) <-> A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. X ((PCw) < z -> ((F` P)D(F` w)) < y))))))
133	132	pm5.32d 709	. 2 |- ((C e. Met /\ D e. Met /\ P e. X) -> ((F:X-->Y /\ A.v e. K ((F` P) e. v -> E.u e. J (P e. u /\ (F"u) C_ v))) <-> (F:X-->Y /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. X ((PCw) < z -> ((F` P)D(F` w)) < y))))))
134	15, 21, 133	3bitrd 603	1 |- ((C e. Met /\ D e. Met /\ P e. X) -> (F e. ((J CnP K)` P) <-> (F:X-->Y /\ A.y e. RR (0 < y -> E.z e. RR (0 < z /\ A.w e. X ((PCw) < z -> ((F` P)D(F` w)) < y))))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   C_ wss 2593  U.cuni 3177   class class class wbr 3338  `'ccnv 3985  dom cdm 3986  "cima 3989  Fun wfun 3992  -->wf 3994  ` cfv 3998  (class class class)co 4884  RRcr 6385  0cc0 6386   < clt 6653  Topctop 8857   CnP ccnp 9029  Metcme 9066   ball cbl 9068  Opencopn 9069

This theorem is referenced by:  metcnp2 9166  metcn 9167  metcnpi 9168  metcnp3 9174

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-top 8861  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073

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