Theorem blfps 15433

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 3333	. . . . . 6 |- { y e. X | ( x D y ) < r } C_ X
2		psmetrel 15346	. . . . . . . 8 |- Rel PsMet
3		relelfvdm 5722	. . . . . . . 8 |- ( ( Rel PsMet /\ D e. (PsMet ` X ) ) -> X e. dom PsMet )
4	2, 3	mpan 428	. . . . . . 7 |- ( D e. (PsMet ` X ) -> X e. dom PsMet )
5		elpw2g 4287	. . . . . . 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 2631	. . 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 2238	. . . 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 6427	. . 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 15409	. . 3 |- ( D e. (PsMet ` X ) -> ( ball ` D ) = ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) )
14	13	feq1d 5515	. 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 2209   A.wral 2528   {crab 2532   C_ wss 3220   ~Pcpw 3685   class class class wbr 4125   X. cxp 4767   dom cdm 4769   Rel wrel 4774   -->wf 5368   ` cfv 5372  (class class class)co 6075   e. cmpo 6077   RR*cxr 8349   < clt 8350  PsMetcpsmet 14844   ballcbl 14847

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-pnf 8352  df-mnf 8353  df-xr 8354  df-psmet 14852  df-bl 14855

This theorem is referenced by:  blrnps  15435  blelrnps  15443  unirnblps  15446