Theorem blfvalps 22188

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 19741	. . 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 11	. 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 5324	. . . . 5 |- ( d = D -> dom d = dom D )
4	3	dmeqd 5326	. . . 4 |- ( d = D -> dom dom d = dom dom D )
5		psmetdmdm 22110	. . . . 5 |- ( D e. (PsMet ` X ) -> X = dom dom D )
6	5	eqcomd 2628	. . . 4 |- ( D e. (PsMet ` X ) -> dom dom D = X )
7	4, 6	sylan9eqr 2678	. . 3 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> dom dom d = X )
8		eqidd 2623	. . 3 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> RR* = RR* )
9		simpr 477	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> d = D )
10	9	oveqd 6667	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( x d y ) = ( x D y ) )
11	10	breq1d 4663	. . . 4 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( ( x d y ) < r <-> ( x D y ) < r ) )
12	7, 11	rabeqbidv 3195	. . 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	mpt2eq123dv 6717	. 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 3212	. 2 |- ( D e. (PsMet ` X ) -> D e. _V )
15		ssrab2 3687	. . . . . 6 |- { y e. X | ( x D y ) < r } C_ X
16		elfvdm 6220	. . . . . . . 8 |- ( D e. (PsMet ` X ) -> X e. dom PsMet )
17	16	adantr 481	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ r e. RR* ) ) -> X e. dom PsMet )
18		elpw2g 4827	. . . . . . 7 |- ( X e. dom PsMet -> ( { y e. X | ( x D y ) < r } e. ~P X <-> { y e. X | ( x D y ) < r } C_ X ) )
19	17, 18	syl 17	. . . . . 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 ) )
20	15, 19	mpbiri 248	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ r e. RR* ) ) -> { y e. X | ( x D y ) < r } e. ~P X )
21	20	ralrimivva 2971	. . . 4 |- ( D e. (PsMet ` X ) -> A. x e. X A. r e. RR* { y e. X | ( x D y ) < r } e. ~P X )
22		eqid 2622	. . . . 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 } )
23	22	fmpt2 7237	. . . 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 )
24	21, 23	sylib 208	. . 3 |- ( D e. (PsMet ` X ) -> ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) : ( X X. RR* ) --> ~P X )
25		xrex 11829	. . . 4 |- RR* e. _V
26		xpexg 6960	. . . 4 |- ( ( X e. dom PsMet /\ RR* e. _V ) -> ( X X. RR* ) e. _V )
27	16, 25, 26	sylancl 694	. . 3 |- ( D e. (PsMet ` X ) -> ( X X. RR* ) e. _V )
28		pwexg 4850	. . . 4 |- ( X e. dom PsMet -> ~P X e. _V )
29	16, 28	syl 17	. . 3 |- ( D e. (PsMet ` X ) -> ~P X e. _V )
30		fex2 7121	. . 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 )
31	24, 27, 29, 30	syl3anc 1326	. 2 |- ( D e. (PsMet ` X ) -> ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) e. _V )
32	2, 13, 14, 31	fvmptd 6288	1 |- ( D e. (PsMet ` X ) -> ( ball ` D ) = ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RR*cxr 10073   < clt 10074  PsMetcpsmet 19730   ballcbl 19733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-xr 10078  df-psmet 19738  df-bl 19741

This theorem is referenced by:  blfval  22189  blvalps  22190  blfps  22211