Theorem blfps 15220

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 3313	. . . . . 6 |- { y e. X | ( x D y ) < r } C_ X
2		psmetrel 15133	. . . . . . . 8 |- Rel PsMet
3		relelfvdm 5680	. . . . . . . 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 4251	. . . . . . 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 2614	. . 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 2231	. . . 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 6375	. . 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 15196	. . 3 |- ( D e. (PsMet ` X ) -> ( ball ` D ) = ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) )
14	13	feq1d 5476	. 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 2202   A.wral 2511   {crab 2515   C_ wss 3201   ~Pcpw 3656   class class class wbr 4093   X. cxp 4729   dom cdm 4731   Rel wrel 4736   -->wf 5329   ` cfv 5333  (class class class)co 6028   e. cmpo 6030   RR*cxr 8272   < clt 8273  PsMetcpsmet 14631   ballcbl 14634

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-pnf 8275  df-mnf 8276  df-xr 8277  df-psmet 14639  df-bl 14642

This theorem is referenced by:  blrnps  15222  blelrnps  15230  unirnblps  15233