Theorem unitg 15086

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 15083	. . . . . 6 |- ( x e. ( topGen ` B ) -> x C_ U. B )
2		velpw 3692	. . . . . 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 3252	. . . 4 |- ( topGen ` B ) C_ ~P U. B
5		sspwuni 4092	. . . 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 15085	. . 3 |- ( B e. V -> B C_ ( topGen ` B ) )
9	8	unissd 3954	. 2 |- ( B e. V -> U. B C_ U. ( topGen ` B ) )
10	7, 9	eqssd 3265	1 |- ( B e. V -> U. ( topGen ` B ) = U. B )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   C_ wss 3220   ~Pcpw 3685   U.cuni 3930   ` cfv 5372   topGenctg 13585

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-topgen 13591

This theorem is referenced by:  tgcl  15088  tgtopon  15090  txtopon  15286  uniretop  15549