Theorem unitg 14927

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 14924	. . . . . 6 |- ( x e. ( topGen ` B ) -> x C_ U. B )
2		velpw 3676	. . . . . 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 3242	. . . 4 |- ( topGen ` B ) C_ ~P U. B
5		sspwuni 4076	. . . 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 14926	. . 3 |- ( B e. V -> B C_ ( topGen ` B ) )
9	8	unissd 3938	. 2 |- ( B e. V -> U. B C_ U. ( topGen ` B ) )
10	7, 9	eqssd 3255	1 |- ( B e. V -> U. ( topGen ` B ) = U. B )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   C_ wss 3211   ~Pcpw 3669   U.cuni 3914   ` cfv 5352   topGenctg 13467

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-topgen 13473

This theorem is referenced by:  tgcl  14929  tgtopon  14931  txtopon  15127  uniretop  15390