Theorem eltg 14220

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 12873	. . 3 |- ( B e. V -> ( topGen ` B ) = { x | x C_ U. ( B i^i ~P x ) } )
2	1	eleq2d 2263	. 2 |- ( B e. V -> ( A e. ( topGen ` B ) <-> A e. { x | x C_ U. ( B i^i ~P x ) } ) )
3		elex 2771	. . . 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 4165	. . . . . 6 |- ( B e. V -> ( B i^i ~P A ) e. _V )
6		uniexg 4470	. . . . . 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 4168	. . . . 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 3604	. . . . . . 7 |- ( x = A -> ~P x = ~P A )
13	12	ineq2d 3360	. . . . . 6 |- ( x = A -> ( B i^i ~P x ) = ( B i^i ~P A ) )
14	13	unieqd 3846	. . . . 5 |- ( x = A -> U. ( B i^i ~P x ) = U. ( B i^i ~P A ) )
15	11, 14	sseq12d 3210	. . . 4 |- ( x = A -> ( x C_ U. ( B i^i ~P x ) <-> A C_ U. ( B i^i ~P A ) ) )
16	15	elabg 2906	. . 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 2164   {cab 2179   _Vcvv 2760   i^i cin 3152   C_ wss 3153   ~Pcpw 3601   U.cuni 3835   ` cfv 5254   topGenctg 12865

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-topgen 12871

This theorem is referenced by:  eltg4i  14223  eltg3i  14224  bastg  14229  tgss  14231  eltop  14237