Theorem blbas 15534

Description: The balls of a metric space form a basis for a topology. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 15-Jan-2014.)

Assertion

Ref	Expression
blbas	|- ( D e. ( *Met ` X ) -> ran ( ball ` D ) e. TopBases )

Proof of Theorem blbas

Dummy variables x r b y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		blin2 15533	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) /\ ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) ) -> E. r e. RR+ ( z ( ball ` D ) r ) C_ ( x i^i y ) )
2		simpll 531	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) /\ ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) ) -> D e. ( *Met ` X ) )
3		elinel1 3415	. . . . . . . . . 10 |- ( z e. ( x i^i y ) -> z e. x )
4		elunii 3940	. . . . . . . . . 10 |- ( ( z e. x /\ x e. ran ( ball ` D ) ) -> z e. U. ran ( ball ` D ) )
5	3, 4	sylan 283	. . . . . . . . 9 |- ( ( z e. ( x i^i y ) /\ x e. ran ( ball ` D ) ) -> z e. U. ran ( ball ` D ) )
6	5	ad2ant2lr 514	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) /\ ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) ) -> z e. U. ran ( ball ` D ) )
7		unirnbl 15524	. . . . . . . . 9 |- ( D e. ( *Met ` X ) -> U. ran ( ball ` D ) = X )
8	7	ad2antrr 492	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) /\ ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) ) -> U. ran ( ball ` D ) = X )
9	6, 8	eleqtrd 2317	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) /\ ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) ) -> z e. X )
10		blssex 15531	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ z e. X ) -> ( E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) <-> E. r e. RR+ ( z ( ball ` D ) r ) C_ ( x i^i y ) ) )
11	2, 9, 10	syl2anc 415	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) /\ ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) ) -> ( E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) <-> E. r e. RR+ ( z ( ball ` D ) r ) C_ ( x i^i y ) ) )
12	1, 11	mpbird 167	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) /\ ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) ) -> E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) )
13	12	ex 115	. . . 4 |- ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) -> ( ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) -> E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) ) )
14	13	ralrimdva 2630	. . 3 |- ( D e. ( *Met ` X ) -> ( ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) -> A. z e. ( x i^i y ) E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) ) )
15	14	ralrimivv 2631	. 2 |- ( D e. ( *Met ` X ) -> A. x e. ran ( ball ` D ) A. y e. ran ( ball ` D ) A. z e. ( x i^i y ) E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) )
16		blex 15488	. . 3 |- ( D e. ( *Met ` X ) -> ( ball ` D ) e. _V )
17		rnexg 5047	. . 3 |- ( ( ball ` D ) e. _V -> ran ( ball ` D ) e. _V )
18		isbasis2g 15146	. . 3 |- ( ran ( ball ` D ) e. _V -> ( ran ( ball ` D ) e. TopBases <-> A. x e. ran ( ball ` D ) A. y e. ran ( ball ` D ) A. z e. ( x i^i y ) E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) ) )
19	16, 17, 18	3syl 17	. 2 |- ( D e. ( *Met ` X ) -> ( ran ( ball ` D ) e. TopBases <-> A. x e. ran ( ball ` D ) A. y e. ran ( ball ` D ) A. z e. ( x i^i y ) E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) ) )
20	15, 19	mpbird 167	1 |- ( D e. ( *Met ` X ) -> ran ( ball ` D ) e. TopBases )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   _Vcvv 2821   i^i cin 3219   C_ wss 3220   U.cuni 3935   ran crn 4775   ` cfv 5377  (class class class)co 6085   RR+crp 10054   *Metcxmet 14873   ballcbl 14875   TopBasesctb 15143

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-map 6924  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-xneg 10174  df-xadd 10175  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-psmet 14880  df-xmet 14881  df-bl 14883  df-bases 15144

This theorem is used by:  mopnval  15543  mopntopon  15544  elmopn  15547  blssopn  15586  metss  15595