Theorem unitg 13647

Description: The topology generated by a basis B is a topology on U. B. Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class TopBases completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof shortened by OpenAI, 30-Mar-2020.)

Assertion

Ref	Expression
unitg	|- ( B e. V -> U. ( topGen ` B ) = U. B )

Proof of Theorem unitg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tg1 13644	. . . . . 6 |- ( x e. ( topGen ` B ) -> x C_ U. B )
2		velpw 3584	. . . . . 6 |- ( x e. ~P U. B <-> x C_ U. B )
3	1, 2	sylibr 134	. . . . 5 |- ( x e. ( topGen ` B ) -> x e. ~P U. B )
4	3	ssriv 3161	. . . 4 |- ( topGen ` B ) C_ ~P U. B
5		sspwuni 3973	. . . 4 |- ( ( topGen ` B ) C_ ~P U. B <-> U. ( topGen ` B ) C_ U. B )
6	4, 5	mpbi 145	. . 3 |- U. ( topGen ` B ) C_ U. B
7	6	a1i 9	. 2 |- ( B e. V -> U. ( topGen ` B ) C_ U. B )
8		bastg 13646	. . 3 |- ( B e. V -> B C_ ( topGen ` B ) )
9	8	unissd 3835	. 2 |- ( B e. V -> U. B C_ U. ( topGen ` B ) )
10	7, 9	eqssd 3174	1 |- ( B e. V -> U. ( topGen ` B ) = U. B )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   C_ wss 3131   ~Pcpw 3577   U.cuni 3811   ` cfv 5218   topGenctg 12708

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-topgen 12714

This theorem is referenced by:  tgcl  13649  tgtopon  13651  txtopon  13847  uniretop  14110