Theorem unitg 14785

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 14782	. . . . . 6 |- ( x e. ( topGen ` B ) -> x C_ U. B )
2		velpw 3659	. . . . . 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 3231	. . . 4 |- ( topGen ` B ) C_ ~P U. B
5		sspwuni 4055	. . . 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 14784	. . 3 |- ( B e. V -> B C_ ( topGen ` B ) )
9	8	unissd 3917	. 2 |- ( B e. V -> U. B C_ U. ( topGen ` B ) )
10	7, 9	eqssd 3244	1 |- ( B e. V -> U. ( topGen ` B ) = U. B )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   C_ wss 3200   ~Pcpw 3652   U.cuni 3893   ` cfv 5326   topGenctg 13336

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-topgen 13342

This theorem is referenced by:  tgcl  14787  tgtopon  14789  txtopon  14985  uniretop  15248