Theorem eltg 14466

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 13036	. . 3 |- ( B e. V -> ( topGen ` B ) = { x | x C_ U. ( B i^i ~P x ) } )
2	1	eleq2d 2274	. 2 |- ( B e. V -> ( A e. ( topGen ` B ) <-> A e. { x | x C_ U. ( B i^i ~P x ) } ) )
3		elex 2782	. . . 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 4179	. . . . . 6 |- ( B e. V -> ( B i^i ~P A ) e. _V )
6		uniexg 4485	. . . . . 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 4182	. . . . 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 3618	. . . . . . 7 |- ( x = A -> ~P x = ~P A )
13	12	ineq2d 3373	. . . . . 6 |- ( x = A -> ( B i^i ~P x ) = ( B i^i ~P A ) )
14	13	unieqd 3860	. . . . 5 |- ( x = A -> U. ( B i^i ~P x ) = U. ( B i^i ~P A ) )
15	11, 14	sseq12d 3223	. . . 4 |- ( x = A -> ( x C_ U. ( B i^i ~P x ) <-> A C_ U. ( B i^i ~P A ) ) )
16	15	elabg 2918	. . 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 1372   e. wcel 2175   {cab 2190   _Vcvv 2771   i^i cin 3164   C_ wss 3165   ~Pcpw 3615   U.cuni 3849   ` cfv 5270   topGenctg 13028

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278  df-topgen 13034

This theorem is referenced by:  eltg4i  14469  eltg3i  14470  bastg  14475  tgss  14477  eltop  14483