Theorem xmetec 22239

Description: The equivalence classes under the finite separation equivalence relation are infinity balls. Thus, by erdisj 7794, infinity balls are either identical or disjoint, quite unlike the usual situation with Euclidean balls which admit many kinds of overlap. (Contributed by Mario Carneiro, 24-Aug-2015.)

Hypothesis

Ref	Expression
xmeter.1	|- .~ = ( `' D " RR )

Assertion

Ref	Expression
xmetec	|- ( ( D e. ( *Met ` X ) /\ P e. X ) -> [ P ] .~ = ( P ( ball ` D ) +oo ) )

Proof of Theorem xmetec

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xmeter.1	. . . . 5 |- .~ = ( `' D " RR )
2	1	xmeterval 22237	. . . 4 |- ( D e. ( *Met ` X ) -> ( P .~ x <-> ( P e. X /\ x e. X /\ ( P D x ) e. RR ) ) )
3		3anass 1042	. . . . 5 |- ( ( P e. X /\ x e. X /\ ( P D x ) e. RR ) <-> ( P e. X /\ ( x e. X /\ ( P D x ) e. RR ) ) )
4	3	baib 944	. . . 4 |- ( P e. X -> ( ( P e. X /\ x e. X /\ ( P D x ) e. RR ) <-> ( x e. X /\ ( P D x ) e. RR ) ) )
5	2, 4	sylan9bb 736	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( P .~ x <-> ( x e. X /\ ( P D x ) e. RR ) ) )
6		vex 3203	. . . . 5 |- x e. _V
7	6	a1i 11	. . . 4 |- ( D e. ( *Met ` X ) -> x e. _V )
8		elecg 7785	. . . 4 |- ( ( x e. _V /\ P e. X ) -> ( x e. [ P ] .~ <-> P .~ x ) )
9	7, 8	sylan 488	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( x e. [ P ] .~ <-> P .~ x ) )
10		xblpnf 22201	. . 3 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( x e. ( P ( ball ` D ) +oo ) <-> ( x e. X /\ ( P D x ) e. RR ) ) )
11	5, 9, 10	3bitr4d 300	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( x e. [ P ] .~ <-> x e. ( P ( ball ` D ) +oo ) ) )
12	11	eqrdv 2620	1 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> [ P ] .~ = ( P ( ball ` D ) +oo ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   [cec 7740   RRcr 9935   +oocpnf 10071   *Metcxmt 19731   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  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-po 5035  df-so 5036  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-ec 7744  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-psmet 19738  df-xmet 19739  df-bl 19741

This theorem is referenced by:  blssec  22240  blpnfctr  22241