Theorem tgval 20759

Description: The topology generated by a basis. See also tgval2 20760 and tgval3 20767. (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 3212	. 2 |- ( B e. V -> B e. _V )
2		uniexg 6955	. . 3 |- ( B e. V -> U. B e. _V )
3		abssexg 4851	. . 3 |- ( U. B e. _V -> { x | ( x C_ U. B /\ x C_ U. ~P x ) } e. _V )
4		uniin 4457	. . . . . . 7 |- U. ( B i^i ~P x ) C_ ( U. B i^i U. ~P x )
5		sstr 3611	. . . . . . 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 707	. . . . . 6 |- ( x C_ U. ( B i^i ~P x ) -> x C_ ( U. B i^i U. ~P x ) )
7		ssin 3835	. . . . . 6 |- ( ( x C_ U. B /\ x C_ U. ~P x ) <-> x C_ ( U. B i^i U. ~P x ) )
8	6, 7	sylibr 224	. . . . 5 |- ( x C_ U. ( B i^i ~P x ) -> ( x C_ U. B /\ x C_ U. ~P x ) )
9	8	ss2abi 3674	. . . 4 |- { x | x C_ U. ( B i^i ~P x ) } C_ { x | ( x C_ U. B /\ x C_ U. ~P x ) }
10		ssexg 4804	. . . 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 706	. . 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 18	. 2 |- ( B e. V -> { x | x C_ U. ( B i^i ~P x ) } e. _V )
13		ineq1 3807	. . . . . 6 |- ( y = B -> ( y i^i ~P x ) = ( B i^i ~P x ) )
14	13	unieqd 4446	. . . . 5 |- ( y = B -> U. ( y i^i ~P x ) = U. ( B i^i ~P x ) )
15	14	sseq2d 3633	. . . 4 |- ( y = B -> ( x C_ U. ( y i^i ~P x ) <-> x C_ U. ( B i^i ~P x ) ) )
16	15	abbidv 2741	. . 3 |- ( y = B -> { x | x C_ U. ( y i^i ~P x ) } = { x | x C_ U. ( B i^i ~P x ) } )
17		df-topgen 16104	. . 3 |- topGen = ( y e. _V |-> { x | x C_ U. ( y i^i ~P x ) } )
18	16, 17	fvmptg 6280	. 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 693	1 |- ( B e. V -> ( topGen ` B ) = { x | x C_ U. ( B i^i ~P x ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   ` cfv 5888   topGenctg 16098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-topgen 16104

This theorem is referenced by:  tgval2  20760  eltg  20761  tgdif0  20796