Theorem eltg 12260

Description: Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Assertion

Ref	Expression
eltg	|- ( B e. V -> ( A e. ( topGen ` B ) <-> A C_ U. ( B i^i ~P A ) ) )

Proof of Theorem eltg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgval 12257	. . 3 |- ( B e. V -> ( topGen ` B ) = { x | x C_ U. ( B i^i ~P x ) } )
2	1	eleq2d 2210	. 2 |- ( B e. V -> ( A e. ( topGen ` B ) <-> A e. { x | x C_ U. ( B i^i ~P x ) } ) )
3		elex 2700	. . . 4 |- ( A e. { x | x C_ U. ( B i^i ~P x ) } -> A e. _V )
4	3	adantl 275	. . 3 |- ( ( B e. V /\ A e. { x | x C_ U. ( B i^i ~P x ) } ) -> A e. _V )
5		inex1g 4072	. . . . . 6 |- ( B e. V -> ( B i^i ~P A ) e. _V )
6		uniexg 4369	. . . . . 6 |- ( ( B i^i ~P A ) e. _V -> U. ( B i^i ~P A ) e. _V )
7	5, 6	syl 14	. . . . 5 |- ( B e. V -> U. ( B i^i ~P A ) e. _V )
8		ssexg 4075	. . . . 5 |- ( ( A C_ U. ( B i^i ~P A ) /\ U. ( B i^i ~P A ) e. _V ) -> A e. _V )
9	7, 8	sylan2 284	. . . 4 |- ( ( A C_ U. ( B i^i ~P A ) /\ B e. V ) -> A e. _V )
10	9	ancoms 266	. . 3 |- ( ( B e. V /\ A C_ U. ( B i^i ~P A ) ) -> A e. _V )
11		id 19	. . . . 5 |- ( x = A -> x = A )
12		pweq 3518	. . . . . . 7 |- ( x = A -> ~P x = ~P A )
13	12	ineq2d 3282	. . . . . 6 |- ( x = A -> ( B i^i ~P x ) = ( B i^i ~P A ) )
14	13	unieqd 3755	. . . . 5 |- ( x = A -> U. ( B i^i ~P x ) = U. ( B i^i ~P A ) )
15	11, 14	sseq12d 3133	. . . 4 |- ( x = A -> ( x C_ U. ( B i^i ~P x ) <-> A C_ U. ( B i^i ~P A ) ) )
16	15	elabg 2834	. . 3 |- ( A e. _V -> ( A e. { x | x C_ U. ( B i^i ~P x ) } <-> A C_ U. ( B i^i ~P A ) ) )
17	4, 10, 16	pm5.21nd 902	. 2 |- ( B e. V -> ( A e. { x | x C_ U. ( B i^i ~P x ) } <-> A C_ U. ( B i^i ~P A ) ) )
18	2, 17	bitrd 187	1 |- ( B e. V -> ( A e. ( topGen ` B ) <-> A C_ U. ( B i^i ~P A ) ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   e. wcel 1481   {cab 2126   _Vcvv 2689   i^i cin 3075   C_ wss 3076   ~Pcpw 3515   U.cuni 3744   ` cfv 5131   topGenctg 12174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-topgen 12180

This theorem is referenced by:  eltg4i  12263  eltg3i  12264  bastg  12269  tgss  12271  eltop  12277