Theorem eltg 14029

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 12770	. . 3 |- ( B e. V -> ( topGen ` B ) = { x | x C_ U. ( B i^i ~P x ) } )
2	1	eleq2d 2259	. 2 |- ( B e. V -> ( A e. ( topGen ` B ) <-> A e. { x | x C_ U. ( B i^i ~P x ) } ) )
3		elex 2763	. . . 4 |- ( A e. { x | x C_ U. ( B i^i ~P x ) } -> A e. _V )
4	3	adantl 277	. . 3 |- ( ( B e. V /\ A e. { x | x C_ U. ( B i^i ~P x ) } ) -> A e. _V )
5		inex1g 4154	. . . . . 6 |- ( B e. V -> ( B i^i ~P A ) e. _V )
6		uniexg 4457	. . . . . 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 4157	. . . . 5 |- ( ( A C_ U. ( B i^i ~P A ) /\ U. ( B i^i ~P A ) e. _V ) -> A e. _V )
9	7, 8	sylan2 286	. . . 4 |- ( ( A C_ U. ( B i^i ~P A ) /\ B e. V ) -> A e. _V )
10	9	ancoms 268	. . 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 3593	. . . . . . 7 |- ( x = A -> ~P x = ~P A )
13	12	ineq2d 3351	. . . . . 6 |- ( x = A -> ( B i^i ~P x ) = ( B i^i ~P A ) )
14	13	unieqd 3835	. . . . 5 |- ( x = A -> U. ( B i^i ~P x ) = U. ( B i^i ~P A ) )
15	11, 14	sseq12d 3201	. . . 4 |- ( x = A -> ( x C_ U. ( B i^i ~P x ) <-> A C_ U. ( B i^i ~P A ) ) )
16	15	elabg 2898	. . 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 917	. 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 188	1 |- ( B e. V -> ( A e. ( topGen ` B ) <-> A C_ U. ( B i^i ~P A ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2160   {cab 2175   _Vcvv 2752   i^i cin 3143   C_ wss 3144   ~Pcpw 3590   U.cuni 3824   ` cfv 5235   topGenctg 12762

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-topgen 12768

This theorem is referenced by:  eltg4i  14032  eltg3i  14033  bastg  14038  tgss  14040  eltop  14046