Theorem bastop1 14555

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 14535	. . . . 5 |- ( ( J e. Top /\ B C_ J ) -> ( topGen ` B ) C_ ( topGen ` J ) )
2		tgtop 14540	. . . . . 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 3231	. . . 4 |- ( ( J e. Top /\ B C_ J ) -> ( topGen ` B ) C_ J )
5		eqss 3208	. . . . 5 |- ( ( topGen ` B ) = J <-> ( ( topGen ` B ) C_ J /\ J C_ ( topGen ` B ) ) )
6	5	baib 921	. . . 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 3182	. . 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 4183	. . . . 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 14529	. . . 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 2506	. 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 1373   E.wex 1515   e. wcel 2176   A.wral 2484   _Vcvv 2772   C_ wss 3166   U.cuni 3850   ` cfv 5271   topGenctg 13086   Topctop 14469

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-topgen 13092  df-top 14470

This theorem is referenced by:  bastop2  14556