Theorem bastop1 14670

Description: A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom " ( topGen ` B ) = J " to express " B is a basis for topology J " since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)

Assertion

Ref	Expression
bastop1	|- ( ( J e. Top /\ B C_ J ) -> ( ( topGen ` B ) = J <-> A. x e. J E. y ( y C_ B /\ x = U. y ) ) )

Distinct variable groups:   x, y, B   x, J, y

Proof of Theorem bastop1

Step	Hyp	Ref	Expression
1		tgss 14650	. . . . 5 |- ( ( J e. Top /\ B C_ J ) -> ( topGen ` B ) C_ ( topGen ` J ) )
2		tgtop 14655	. . . . . 6 |- ( J e. Top -> ( topGen ` J ) = J )
3	2	adantr 276	. . . . 5 |- ( ( J e. Top /\ B C_ J ) -> ( topGen ` J ) = J )
4	1, 3	sseqtrd 3239	. . . 4 |- ( ( J e. Top /\ B C_ J ) -> ( topGen ` B ) C_ J )
5		eqss 3216	. . . . 5 |- ( ( topGen ` B ) = J <-> ( ( topGen ` B ) C_ J /\ J C_ ( topGen ` B ) ) )
6	5	baib 921	. . . 4 |- ( ( topGen ` B ) C_ J -> ( ( topGen ` B ) = J <-> J C_ ( topGen ` B ) ) )
7	4, 6	syl 14	. . 3 |- ( ( J e. Top /\ B C_ J ) -> ( ( topGen ` B ) = J <-> J C_ ( topGen ` B ) ) )
8		dfss3 3190	. . 3 |- ( J C_ ( topGen ` B ) <-> A. x e. J x e. ( topGen ` B ) )
9	7, 8	bitrdi 196	. 2 |- ( ( J e. Top /\ B C_ J ) -> ( ( topGen ` B ) = J <-> A. x e. J x e. ( topGen ` B ) ) )
10		ssexg 4199	. . . . 5 |- ( ( B C_ J /\ J e. Top ) -> B e. _V )
11	10	ancoms 268	. . . 4 |- ( ( J e. Top /\ B C_ J ) -> B e. _V )
12		eltg3 14644	. . . 4 |- ( B e. _V -> ( x e. ( topGen ` B ) <-> E. y ( y C_ B /\ x = U. y ) ) )
13	11, 12	syl 14	. . 3 |- ( ( J e. Top /\ B C_ J ) -> ( x e. ( topGen ` B ) <-> E. y ( y C_ B /\ x = U. y ) ) )
14	13	ralbidv 2508	. 2 |- ( ( J e. Top /\ B C_ J ) -> ( A. x e. J x e. ( topGen ` B ) <-> A. x e. J E. y ( y C_ B /\ x = U. y ) ) )
15	9, 14	bitrd 188	1 |- ( ( J e. Top /\ B C_ J ) -> ( ( topGen ` B ) = J <-> A. x e. J E. y ( y C_ B /\ x = U. y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2178   A.wral 2486   _Vcvv 2776   C_ wss 3174   U.cuni 3864   ` cfv 5290   topGenctg 13201   Topctop 14584

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-topgen 13207  df-top 14585

This theorem is referenced by:  bastop2  14671