Theorem eltg 11919

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 11916	. . 3 |- ( B e. V -> ( topGen ` B ) = { x | x C_ U. ( B i^i ~P x ) } )
2	1	eleq2d 2164	. 2 |- ( B e. V -> ( A e. ( topGen ` B ) <-> A e. { x | x C_ U. ( B i^i ~P x ) } ) )
3		elex 2644	. . . 4 |- ( A e. { x | x C_ U. ( B i^i ~P x ) } -> A e. _V )
4	3	adantl 272	. . 3 |- ( ( B e. V /\ A e. { x | x C_ U. ( B i^i ~P x ) } ) -> A e. _V )
5		inex1g 3996	. . . . . 6 |- ( B e. V -> ( B i^i ~P A ) e. _V )
6		uniexg 4290	. . . . . 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 3999	. . . . 5 |- ( ( A C_ U. ( B i^i ~P A ) /\ U. ( B i^i ~P A ) e. _V ) -> A e. _V )
9	7, 8	sylan2 281	. . . 4 |- ( ( A C_ U. ( B i^i ~P A ) /\ B e. V ) -> A e. _V )
10	9	ancoms 265	. . 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 3452	. . . . . . 7 |- ( x = A -> ~P x = ~P A )
13	12	ineq2d 3216	. . . . . 6 |- ( x = A -> ( B i^i ~P x ) = ( B i^i ~P A ) )
14	13	unieqd 3686	. . . . 5 |- ( x = A -> U. ( B i^i ~P x ) = U. ( B i^i ~P A ) )
15	11, 14	sseq12d 3070	. . . 4 |- ( x = A -> ( x C_ U. ( B i^i ~P x ) <-> A C_ U. ( B i^i ~P A ) ) )
16	15	elabg 2775	. . 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 866	. 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 1296   e. wcel 1445   {cab 2081   _Vcvv 2633   i^i cin 3012   C_ wss 3013   ~Pcpw 3449   U.cuni 3675   ` cfv 5049   topGenctg 11834

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-topgen 11840

This theorem is referenced by:  eltg4i  11922  eltg3i  11923  bastg  11928  tgss  11930  eltop  11936