Theorem blrnps 14138

Description: Membership in the range of the ball function. Note that ran ( ball ` D ) is the collection of all balls for metric D. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)

Assertion

Ref	Expression
blrnps	|- ( D e. (PsMet ` X ) -> ( A e. ran ( ball ` D ) <-> E. x e. X E. r e. RR* A = ( x ( ball ` D ) r ) ) )

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

Proof of Theorem blrnps

Step	Hyp	Ref	Expression
1		blfps 14136	. 2 |- ( D e. (PsMet ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )
2		ffn 5377	. 2 |- ( ( ball ` D ) : ( X X. RR* ) --> ~P X -> ( ball ` D ) Fn ( X X. RR* ) )
3		ovelrn 6036	. 2 |- ( ( ball ` D ) Fn ( X X. RR* ) -> ( A e. ran ( ball ` D ) <-> E. x e. X E. r e. RR* A = ( x ( ball ` D ) r ) ) )
4	1, 2, 3	3syl 17	1 |- ( D e. (PsMet ` X ) -> ( A e. ran ( ball ` D ) <-> E. x e. X E. r e. RR* A = ( x ( ball ` D ) r ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1363   e. wcel 2158   E.wrex 2466   ~Pcpw 3587   X. cxp 4636   ran crn 4639   Fn wfn 5223   -->wf 5224   ` cfv 5228  (class class class)co 5888   RR*cxr 8004  PsMetcpsmet 13652   ballcbl 13655

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-map 6663  df-pnf 8007  df-mnf 8008  df-xr 8009  df-psmet 13660  df-bl 13663

This theorem is referenced by:  blssps  14154