Theorem bastop1 14757

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 14737	. . . . 5 |- ( ( J e. Top /\ B C_ J ) -> ( topGen ` B ) C_ ( topGen ` J ) )
2		tgtop 14742	. . . . . 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 3262	. . . 4 |- ( ( J e. Top /\ B C_ J ) -> ( topGen ` B ) C_ J )
5		eqss 3239	. . . . 5 |- ( ( topGen ` B ) = J <-> ( ( topGen ` B ) C_ J /\ J C_ ( topGen ` B ) ) )
6	5	baib 924	. . . 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 3213	. . 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 4223	. . . . 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 14731	. . . 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 2530	. 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 1395   E.wex 1538   e. wcel 2200   A.wral 2508   _Vcvv 2799   C_ wss 3197   U.cuni 3888   ` cfv 5318   topGenctg 13287   Topctop 14671

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-topgen 13293  df-top 14672

This theorem is referenced by:  bastop2  14758