Theorem blfvalps 14705

Description: The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Feb-2018.)

Assertion

Ref	Expression
blfvalps	|- ( D e. (PsMet ` X ) -> ( ball ` D ) = ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) )

Distinct variable groups:   x, r, y, D   X, r, x, y

Proof of Theorem blfvalps

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bl 14178	. . 3 |- ball = ( d e. _V |-> ( x e. dom dom d , r e. RR* |-> { y e. dom dom d | ( x d y ) < r } ) )
2	1	a1i 9	. 2 |- ( D e. (PsMet ` X ) -> ball = ( d e. _V |-> ( x e. dom dom d , r e. RR* |-> { y e. dom dom d | ( x d y ) < r } ) ) )
3		dmeq 4867	. . . . 5 |- ( d = D -> dom d = dom D )
4	3	dmeqd 4869	. . . 4 |- ( d = D -> dom dom d = dom dom D )
5		psmetdmdm 14644	. . . . 5 |- ( D e. (PsMet ` X ) -> X = dom dom D )
6	5	eqcomd 2202	. . . 4 |- ( D e. (PsMet ` X ) -> dom dom D = X )
7	4, 6	sylan9eqr 2251	. . 3 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> dom dom d = X )
8		eqidd 2197	. . 3 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> RR* = RR* )
9		simpr 110	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> d = D )
10	9	oveqd 5942	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( x d y ) = ( x D y ) )
11	10	breq1d 4044	. . . 4 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( ( x d y ) < r <-> ( x D y ) < r ) )
12	7, 11	rabeqbidv 2758	. . 3 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> { y e. dom dom d | ( x d y ) < r } = { y e. X | ( x D y ) < r } )
13	7, 8, 12	mpoeq123dv 5988	. 2 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( x e. dom dom d , r e. RR* |-> { y e. dom dom d | ( x d y ) < r } ) = ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) )
14		elex 2774	. 2 |- ( D e. (PsMet ` X ) -> D e. _V )
15		ssrab2 3269	. . . . . 6 |- { y e. X | ( x D y ) < r } C_ X
16		psmetrel 14642	. . . . . . . . 9 |- Rel PsMet
17		relelfvdm 5593	. . . . . . . . 9 |- ( ( Rel PsMet /\ D e. (PsMet ` X ) ) -> X e. dom PsMet )
18	16, 17	mpan 424	. . . . . . . 8 |- ( D e. (PsMet ` X ) -> X e. dom PsMet )
19	18	adantr 276	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ r e. RR* ) ) -> X e. dom PsMet )
20		elpw2g 4190	. . . . . . 7 |- ( X e. dom PsMet -> ( { y e. X | ( x D y ) < r } e. ~P X <-> { y e. X | ( x D y ) < r } C_ X ) )
21	19, 20	syl 14	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ r e. RR* ) ) -> ( { y e. X | ( x D y ) < r } e. ~P X <-> { y e. X | ( x D y ) < r } C_ X ) )
22	15, 21	mpbiri 168	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ r e. RR* ) ) -> { y e. X | ( x D y ) < r } e. ~P X )
23	22	ralrimivva 2579	. . . 4 |- ( D e. (PsMet ` X ) -> A. x e. X A. r e. RR* { y e. X | ( x D y ) < r } e. ~P X )
24		eqid 2196	. . . . 5 |- ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) = ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } )
25	24	fmpo 6268	. . . 4 |- ( A. x e. X A. r e. RR* { y e. X | ( x D y ) < r } e. ~P X <-> ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) : ( X X. RR* ) --> ~P X )
26	23, 25	sylib 122	. . 3 |- ( D e. (PsMet ` X ) -> ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) : ( X X. RR* ) --> ~P X )
27		xrex 9948	. . . 4 |- RR* e. _V
28		xpexg 4778	. . . 4 |- ( ( X e. dom PsMet /\ RR* e. _V ) -> ( X X. RR* ) e. _V )
29	18, 27, 28	sylancl 413	. . 3 |- ( D e. (PsMet ` X ) -> ( X X. RR* ) e. _V )
30	18	pwexd 4215	. . 3 |- ( D e. (PsMet ` X ) -> ~P X e. _V )
31		fex2 5429	. . 3 |- ( ( ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) : ( X X. RR* ) --> ~P X /\ ( X X. RR* ) e. _V /\ ~P X e. _V ) -> ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) e. _V )
32	26, 29, 30, 31	syl3anc 1249	. 2 |- ( D e. (PsMet ` X ) -> ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) e. _V )
33	2, 13, 14, 32	fvmptd 5645	1 |- ( D e. (PsMet ` X ) -> ( ball ` D ) = ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   {crab 2479   _Vcvv 2763   C_ wss 3157   ~Pcpw 3606   class class class wbr 4034   |-> cmpt 4095   X. cxp 4662   dom cdm 4664   Rel wrel 4669   -->wf 5255   ` cfv 5259  (class class class)co 5925   e. cmpo 5927   RR*cxr 8077   < clt 8078  PsMetcpsmet 14167   ballcbl 14170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-pnf 8080  df-mnf 8081  df-xr 8082  df-psmet 14175  df-bl 14178

This theorem is referenced by:  blfval  14706  blvalps  14708  blfps  14729