Theorem bastop1 14948

Description: A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom " ( topGen ` B ) = J " to express " B is a basis for topology J " since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)

Assertion

Ref	Expression
bastop1	|- ( ( J e. Top /\ B C_ J ) -> ( ( topGen ` B ) = J <-> A. x e. J E. y ( y C_ B /\ x = U. y ) ) )

Distinct variable groups:   x, y, B   x, J, y

Proof of Theorem bastop1

Step	Hyp	Ref	Expression
1		tgss 14928	. . . . 5 |- ( ( J e. Top /\ B C_ J ) -> ( topGen ` B ) C_ ( topGen ` J ) )
2		tgtop 14933	. . . . . 6 |- ( J e. Top -> ( topGen ` J ) = J )
3	2	adantr 276	. . . . 5 |- ( ( J e. Top /\ B C_ J ) -> ( topGen ` J ) = J )
4	1, 3	sseqtrd 3276	. . . 4 |- ( ( J e. Top /\ B C_ J ) -> ( topGen ` B ) C_ J )
5		eqss 3253	. . . . 5 |- ( ( topGen ` B ) = J <-> ( ( topGen ` B ) C_ J /\ J C_ ( topGen ` B ) ) )
6	5	baib 927	. . . 4 |- ( ( topGen ` B ) C_ J -> ( ( topGen ` B ) = J <-> J C_ ( topGen ` B ) ) )
7	4, 6	syl 14	. . 3 |- ( ( J e. Top /\ B C_ J ) -> ( ( topGen ` B ) = J <-> J C_ ( topGen ` B ) ) )
8		dfss3 3227	. . 3 |- ( J C_ ( topGen ` B ) <-> A. x e. J x e. ( topGen ` B ) )
9	7, 8	bitrdi 196	. 2 |- ( ( J e. Top /\ B C_ J ) -> ( ( topGen ` B ) = J <-> A. x e. J x e. ( topGen ` B ) ) )
10		ssexg 4249	. . . . 5 |- ( ( B C_ J /\ J e. Top ) -> B e. _V )
11	10	ancoms 268	. . . 4 |- ( ( J e. Top /\ B C_ J ) -> B e. _V )
12		eltg3 14922	. . . 4 |- ( B e. _V -> ( x e. ( topGen ` B ) <-> E. y ( y C_ B /\ x = U. y ) ) )
13	11, 12	syl 14	. . 3 |- ( ( J e. Top /\ B C_ J ) -> ( x e. ( topGen ` B ) <-> E. y ( y C_ B /\ x = U. y ) ) )
14	13	ralbidv 2542	. 2 |- ( ( J e. Top /\ B C_ J ) -> ( A. x e. J x e. ( topGen ` B ) <-> A. x e. J E. y ( y C_ B /\ x = U. y ) ) )
15	9, 14	bitrd 188	1 |- ( ( J e. Top /\ B C_ J ) -> ( ( topGen ` B ) = J <-> A. x e. J E. y ( y C_ B /\ x = U. y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2203   A.wral 2520   _Vcvv 2813   C_ wss 3211   U.cuni 3914   ` cfv 5352   topGenctg 13467   Topctop 14862

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-topgen 13473  df-top 14863

This theorem is referenced by:  bastop2  14949