Theorem blfvalps 13465

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 13070	. . 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 4820	. . . . 5 |- ( d = D -> dom d = dom D )
4	3	dmeqd 4822	. . . 4 |- ( d = D -> dom dom d = dom dom D )
5		psmetdmdm 13404	. . . . 5 |- ( D e. (PsMet ` X ) -> X = dom dom D )
6	5	eqcomd 2181	. . . 4 |- ( D e. (PsMet ` X ) -> dom dom D = X )
7	4, 6	sylan9eqr 2230	. . 3 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> dom dom d = X )
8		eqidd 2176	. . 3 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> RR* = RR* )
9		simpr 110	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> d = D )
10	9	oveqd 5882	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( x d y ) = ( x D y ) )
11	10	breq1d 4008	. . . 4 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( ( x d y ) < r <-> ( x D y ) < r ) )
12	7, 11	rabeqbidv 2730	. . 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 5927	. 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 2746	. 2 |- ( D e. (PsMet ` X ) -> D e. _V )
15		ssrab2 3238	. . . . . 6 |- { y e. X | ( x D y ) < r } C_ X
16		psmetrel 13402	. . . . . . . . 9 |- Rel PsMet
17		relelfvdm 5539	. . . . . . . . 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 4151	. . . . . . 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 2557	. . . 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 2175	. . . . 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 6192	. . . 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 9827	. . . 4 |- RR* e. _V
28		xpexg 4734	. . . 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 4176	. . 3 |- ( D e. (PsMet ` X ) -> ~P X e. _V )
31		fex2 5376	. . 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 1238	. 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 5589	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 1353   e. wcel 2146   A.wral 2453   {crab 2457   _Vcvv 2735   C_ wss 3127   ~Pcpw 3572   class class class wbr 3998   |-> cmpt 4059   X. cxp 4618   dom cdm 4620   Rel wrel 4625   -->wf 5204   ` cfv 5208  (class class class)co 5865   e. cmpo 5867   RR*cxr 7965   < clt 7966  PsMetcpsmet 13059   ballcbl 13062

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-map 6640  df-pnf 7968  df-mnf 7969  df-xr 7970  df-psmet 13067  df-bl 13070

This theorem is referenced by:  blfval  13466  blvalps  13468  blfps  13489