Theorem tgval 13475

Description: The topology generated by a basis. See also tgval2 14916 and tgval3 14923. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)

Assertion

Ref	Expression
tgval	|- ( B e. V -> ( topGen ` B ) = { x | x C_ U. ( B i^i ~P x ) } )

Distinct variable groups:   x, B   x, V

Proof of Theorem tgval

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2825	. 2 |- ( B e. V -> B e. _V )
2		uniexg 4560	. . 3 |- ( B e. V -> U. B e. _V )
3		abssexg 4295	. . 3 |- ( U. B e. _V -> { x | ( x C_ U. B /\ x C_ U. ~P x ) } e. _V )
4		uniin 3934	. . . . . . 7 |- U. ( B i^i ~P x ) C_ ( U. B i^i U. ~P x )
5		sstr 3246	. . . . . . 7 |- ( ( x C_ U. ( B i^i ~P x ) /\ U. ( B i^i ~P x ) C_ ( U. B i^i U. ~P x ) ) -> x C_ ( U. B i^i U. ~P x ) )
6	4, 5	mpan2 425	. . . . . 6 |- ( x C_ U. ( B i^i ~P x ) -> x C_ ( U. B i^i U. ~P x ) )
7		ssin 3443	. . . . . 6 |- ( ( x C_ U. B /\ x C_ U. ~P x ) <-> x C_ ( U. B i^i U. ~P x ) )
8	6, 7	sylibr 134	. . . . 5 |- ( x C_ U. ( B i^i ~P x ) -> ( x C_ U. B /\ x C_ U. ~P x ) )
9	8	ss2abi 3310	. . . 4 |- { x | x C_ U. ( B i^i ~P x ) } C_ { x | ( x C_ U. B /\ x C_ U. ~P x ) }
10		ssexg 4249	. . . 4 |- ( ( { x | x C_ U. ( B i^i ~P x ) } C_ { x | ( x C_ U. B /\ x C_ U. ~P x ) } /\ { x | ( x C_ U. B /\ x C_ U. ~P x ) } e. _V ) -> { x | x C_ U. ( B i^i ~P x ) } e. _V )
11	9, 10	mpan 424	. . 3 |- ( { x | ( x C_ U. B /\ x C_ U. ~P x ) } e. _V -> { x | x C_ U. ( B i^i ~P x ) } e. _V )
12	2, 3, 11	3syl 17	. 2 |- ( B e. V -> { x | x C_ U. ( B i^i ~P x ) } e. _V )
13		ineq1 3415	. . . . . 6 |- ( y = B -> ( y i^i ~P x ) = ( B i^i ~P x ) )
14	13	unieqd 3925	. . . . 5 |- ( y = B -> U. ( y i^i ~P x ) = U. ( B i^i ~P x ) )
15	14	sseq2d 3268	. . . 4 |- ( y = B -> ( x C_ U. ( y i^i ~P x ) <-> x C_ U. ( B i^i ~P x ) ) )
16	15	abbidv 2352	. . 3 |- ( y = B -> { x | x C_ U. ( y i^i ~P x ) } = { x | x C_ U. ( B i^i ~P x ) } )
17		df-topgen 13473	. . 3 |- topGen = ( y e. _V |-> { x | x C_ U. ( y i^i ~P x ) } )
18	16, 17	fvmptg 5753	. 2 |- ( ( B e. _V /\ { x | x C_ U. ( B i^i ~P x ) } e. _V ) -> ( topGen ` B ) = { x | x C_ U. ( B i^i ~P x ) } )
19	1, 12, 18	syl2anc 411	1 |- ( B e. V -> ( topGen ` B ) = { x | x C_ U. ( B i^i ~P x ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   {cab 2218   _Vcvv 2813   i^i cin 3210   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:  tgvalex  13476  tgval2  14916  eltg  14917