Theorem blfvalps 14972

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 14423	. . 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 9	. 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 4897	. . . . 5 |- ( d = D -> dom d = dom D )
4	3	dmeqd 4899	. . . 4 |- ( d = D -> dom dom d = dom dom D )
5		psmetdmdm 14911	. . . . 5 |- ( D e. (PsMet ` X ) -> X = dom dom D )
6	5	eqcomd 2213	. . . 4 |- ( D e. (PsMet ` X ) -> dom dom D = X )
7	4, 6	sylan9eqr 2262	. . 3 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> dom dom d = X )
8		eqidd 2208	. . 3 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> RR* = RR* )
9		simpr 110	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> d = D )
10	9	oveqd 5984	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( x d y ) = ( x D y ) )
11	10	breq1d 4069	. . . 4 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( ( x d y ) < r <-> ( x D y ) < r ) )
12	7, 11	rabeqbidv 2771	. . 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	mpoeq123dv 6030	. 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 2788	. 2 |- ( D e. (PsMet ` X ) -> D e. _V )
15		ssrab2 3286	. . . . . 6 |- { y e. X | ( x D y ) < r } C_ X
16		psmetrel 14909	. . . . . . . . 9 |- Rel PsMet
17		relelfvdm 5631	. . . . . . . . 9 |- ( ( Rel PsMet /\ D e. (PsMet ` X ) ) -> X e. dom PsMet )
18	16, 17	mpan 424	. . . . . . . 8 |- ( D e. (PsMet ` X ) -> X e. dom PsMet )
19	18	adantr 276	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ r e. RR* ) ) -> X e. dom PsMet )
20		elpw2g 4216	. . . . . . 7 |- ( X e. dom PsMet -> ( { y e. X | ( x D y ) < r } e. ~P X <-> { y e. X | ( x D y ) < r } C_ X ) )
21	19, 20	syl 14	. . . . . 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 ) )
22	15, 21	mpbiri 168	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ r e. RR* ) ) -> { y e. X | ( x D y ) < r } e. ~P X )
23	22	ralrimivva 2590	. . . 4 |- ( D e. (PsMet ` X ) -> A. x e. X A. r e. RR* { y e. X | ( x D y ) < r } e. ~P X )
24		eqid 2207	. . . . 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 } )
25	24	fmpo 6310	. . . 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 )
26	23, 25	sylib 122	. . 3 |- ( D e. (PsMet ` X ) -> ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) : ( X X. RR* ) --> ~P X )
27		xrex 10013	. . . 4 |- RR* e. _V
28		xpexg 4807	. . . 4 |- ( ( X e. dom PsMet /\ RR* e. _V ) -> ( X X. RR* ) e. _V )
29	18, 27, 28	sylancl 413	. . 3 |- ( D e. (PsMet ` X ) -> ( X X. RR* ) e. _V )
30	18	pwexd 4241	. . 3 |- ( D e. (PsMet ` X ) -> ~P X e. _V )
31		fex2 5464	. . 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 )
32	26, 29, 30, 31	syl3anc 1250	. 2 |- ( D e. (PsMet ` X ) -> ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) e. _V )
33	2, 13, 14, 32	fvmptd 5683	1 |- ( D e. (PsMet ` X ) -> ( ball ` D ) = ( x e. X , r e. RR* |-> { y e. X | ( x D y ) < r } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   A.wral 2486   {crab 2490   _Vcvv 2776   C_ wss 3174   ~Pcpw 3626   class class class wbr 4059   |-> cmpt 4121   X. cxp 4691   dom cdm 4693   Rel wrel 4698   -->wf 5286   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   RR*cxr 8141   < clt 8142  PsMetcpsmet 14412   ballcbl 14415

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-pnf 8144  df-mnf 8145  df-xr 8146  df-psmet 14420  df-bl 14423

This theorem is referenced by:  blfval  14973  blvalps  14975  blfps  14996