Theorem blfvalps 13179

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 12784	. . 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 4811	. . . . 5 |- ( d = D -> dom d = dom D )
4	3	dmeqd 4813	. . . 4 |- ( d = D -> dom dom d = dom dom D )
5		psmetdmdm 13118	. . . . 5 |- ( D e. (PsMet ` X ) -> X = dom dom D )
6	5	eqcomd 2176	. . . 4 |- ( D e. (PsMet ` X ) -> dom dom D = X )
7	4, 6	sylan9eqr 2225	. . 3 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> dom dom d = X )
8		eqidd 2171	. . 3 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> RR* = RR* )
9		simpr 109	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> d = D )
10	9	oveqd 5870	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( x d y ) = ( x D y ) )
11	10	breq1d 3999	. . . 4 |- ( ( D e. (PsMet ` X ) /\ d = D ) -> ( ( x d y ) < r <-> ( x D y ) < r ) )
12	7, 11	rabeqbidv 2725	. . 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 5915	. 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 2741	. 2 |- ( D e. (PsMet ` X ) -> D e. _V )
15		ssrab2 3232	. . . . . 6 |- { y e. X | ( x D y ) < r } C_ X
16		psmetrel 13116	. . . . . . . . 9 |- Rel PsMet
17		relelfvdm 5528	. . . . . . . . 9 |- ( ( Rel PsMet /\ D e. (PsMet ` X ) ) -> X e. dom PsMet )
18	16, 17	mpan 422	. . . . . . . 8 |- ( D e. (PsMet ` X ) -> X e. dom PsMet )
19	18	adantr 274	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ r e. RR* ) ) -> X e. dom PsMet )
20		elpw2g 4142	. . . . . . 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 167	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( x e. X /\ r e. RR* ) ) -> { y e. X | ( x D y ) < r } e. ~P X )
23	22	ralrimivva 2552	. . . 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 2170	. . . . 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 6180	. . . 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 121	. . 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 9813	. . . 4 |- RR* e. _V
28		xpexg 4725	. . . 4 |- ( ( X e. dom PsMet /\ RR* e. _V ) -> ( X X. RR* ) e. _V )
29	18, 27, 28	sylancl 411	. . 3 |- ( D e. (PsMet ` X ) -> ( X X. RR* ) e. _V )
30	18	pwexd 4167	. . 3 |- ( D e. (PsMet ` X ) -> ~P X e. _V )
31		fex2 5366	. . 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 1233	. 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 5577	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 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   {crab 2452   _Vcvv 2730   C_ wss 3121   ~Pcpw 3566   class class class wbr 3989   |-> cmpt 4050   X. cxp 4609   dom cdm 4611   Rel wrel 4616   -->wf 5194   ` cfv 5198  (class class class)co 5853   e. cmpo 5855   RR*cxr 7953   < clt 7954  PsMetcpsmet 12773   ballcbl 12776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-pnf 7956  df-mnf 7957  df-xr 7958  df-psmet 12781  df-bl 12784

This theorem is referenced by:  blfval  13180  blvalps  13182  blfps  13203