Theorem blfps 15083

Description: Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)

Assertion

Ref	Expression
blfps	|- ( D e. (PsMet ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )

Proof of Theorem blfps

Dummy variables x r y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3309	. . . . . 6 |- { y e. X | ( x D y ) < r } C_ X
2		psmetrel 14996	. . . . . . . 8 |- Rel PsMet
3		relelfvdm 5659	. . . . . . . 8 |- ( ( Rel PsMet /\ D e. (PsMet ` X ) ) -> X e. dom PsMet )
4	2, 3	mpan 424	. . . . . . 7 |- ( D e. (PsMet ` X ) -> X e. dom PsMet )
5		elpw2g 4240	. . . . . . 7 |- ( X e. dom PsMet -> ( { y e. X | ( x D y ) < r } e. ~P X <-> { y e. X | ( x D y ) < r } C_ X ) )
6	4, 5	syl 14	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( { y e. X | ( x D y ) < r } e. ~P X <-> { y e. X | ( x D y ) < r } C_ X ) )
7	1, 6	mpbiri 168	. . . . 5 |- ( D e. (PsMet ` X ) -> { y e. X | ( x D y ) < r } e. ~P X )
8	7	a1d 22	. . . 4 |- ( D e. (PsMet ` X ) -> ( ( x e. X /\ r e. RR* ) -> { y e. X | ( x D y ) < r } e. ~P X ) )
9	8	ralrimivv 2611	. . 3 |- ( D e. (PsMet ` X ) -> A. x e. X A. r e. RR* { y e. X | ( x D y ) < r } e. ~P X )
10		eqid 2229	. . . 4 |- ( 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 } )
11	10	fmpo 6347	. . 3 |- ( 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 )
12	9, 11	sylib 122	. 2 |- ( D e. (PsMet ` X ) -> ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) : ( X X. RR* ) --> ~P X )
13		blfvalps 15059	. . 3 |- ( D e. (PsMet ` X ) -> ( ball ` D ) = ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) )
14	13	feq1d 5460	. 2 |- ( D e. (PsMet ` X ) -> ( ( ball ` D ) : ( X X. RR* ) --> ~P X <-> ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) : ( X X. RR* ) --> ~P X ) )
15	12, 14	mpbird 167	1 |- ( D e. (PsMet ` X ) -> ( ball ` D ) : ( X X. RR* ) --> ~P X )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   A.wral 2508   {crab 2512   C_ wss 3197   ~Pcpw 3649   class class class wbr 4083   X. cxp 4717   dom cdm 4719   Rel wrel 4724   -->wf 5314   ` cfv 5318  (class class class)co 6001   e. cmpo 6003   RR*cxr 8180   < clt 8181  PsMetcpsmet 14499   ballcbl 14502

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-map 6797  df-pnf 8183  df-mnf 8184  df-xr 8185  df-psmet 14507  df-bl 14510

This theorem is referenced by:  blrnps  15085  blelrnps  15093  unirnblps  15096