Theorem unitg 12220

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 12217	. . . . . 6 |- ( x e. ( topGen ` B ) -> x C_ U. B )
2		velpw 3512	. . . . . 6 |- ( x e. ~P U. B <-> x C_ U. B )
3	1, 2	sylibr 133	. . . . 5 |- ( x e. ( topGen ` B ) -> x e. ~P U. B )
4	3	ssriv 3096	. . . 4 |- ( topGen ` B ) C_ ~P U. B
5		sspwuni 3892	. . . 4 |- ( ( topGen ` B ) C_ ~P U. B <-> U. ( topGen ` B ) C_ U. B )
6	4, 5	mpbi 144	. . 3 |- U. ( topGen ` B ) C_ U. B
7	6	a1i 9	. 2 |- ( B e. V -> U. ( topGen ` B ) C_ U. B )
8		bastg 12219	. . 3 |- ( B e. V -> B C_ ( topGen ` B ) )
9	8	unissd 3755	. 2 |- ( B e. V -> U. B C_ U. ( topGen ` B ) )
10	7, 9	eqssd 3109	1 |- ( B e. V -> U. ( topGen ` B ) = U. B )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   C_ wss 3066   ~Pcpw 3505   U.cuni 3731   ` cfv 5118   topGenctg 12124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-topgen 12130

This theorem is referenced by:  tgcl  12222  tgtopon  12224  txtopon  12420  uniretop  12683