Theorem tgdom 19994

Description: A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.)

Assertion

Ref	Expression
tgdom	|- ( B e. V -> ( topGen ` B ) ~<_ ~P B )

Proof of Theorem tgdom

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwexg 4587	. 2 |- ( B e. V -> ~P B e. _V )
2		inss1 3652	. . . . 5 |- ( B i^i ~P x ) C_ B
3		vex 3048	. . . . . . . 8 |- x e. _V
4	3	pwex 4586	. . . . . . 7 |- ~P x e. _V
5	4	inex2 4545	. . . . . 6 |- ( B i^i ~P x ) e. _V
6	5	elpw 3957	. . . . 5 |- ( ( B i^i ~P x ) e. ~P B <-> ( B i^i ~P x ) C_ B )
7	2, 6	mpbir 213	. . . 4 |- ( B i^i ~P x ) e. ~P B
8	7	a1i 11	. . 3 |- ( x e. ( topGen ` B ) -> ( B i^i ~P x ) e. ~P B )
9		unieq 4206	. . . . . . 7 |- ( ( B i^i ~P x ) = ( B i^i ~P y ) -> U. ( B i^i ~P x ) = U. ( B i^i ~P y ) )
10	9	adantl 468	. . . . . 6 |- ( ( ( x e. ( topGen ` B ) /\ y e. ( topGen ` B ) ) /\ ( B i^i ~P x ) = ( B i^i ~P y ) ) -> U. ( B i^i ~P x ) = U. ( B i^i ~P y ) )
11		eltg4i 19975	. . . . . . 7 |- ( x e. ( topGen ` B ) -> x = U. ( B i^i ~P x ) )
12	11	ad2antrr 732	. . . . . 6 |- ( ( ( x e. ( topGen ` B ) /\ y e. ( topGen ` B ) ) /\ ( B i^i ~P x ) = ( B i^i ~P y ) ) -> x = U. ( B i^i ~P x ) )
13		eltg4i 19975	. . . . . . 7 |- ( y e. ( topGen ` B ) -> y = U. ( B i^i ~P y ) )
14	13	ad2antlr 733	. . . . . 6 |- ( ( ( x e. ( topGen ` B ) /\ y e. ( topGen ` B ) ) /\ ( B i^i ~P x ) = ( B i^i ~P y ) ) -> y = U. ( B i^i ~P y ) )
15	10, 12, 14	3eqtr4d 2495	. . . . 5 |- ( ( ( x e. ( topGen ` B ) /\ y e. ( topGen ` B ) ) /\ ( B i^i ~P x ) = ( B i^i ~P y ) ) -> x = y )
16	15	ex 436	. . . 4 |- ( ( x e. ( topGen ` B ) /\ y e. ( topGen ` B ) ) -> ( ( B i^i ~P x ) = ( B i^i ~P y ) -> x = y ) )
17		pweq 3954	. . . . 5 |- ( x = y -> ~P x = ~P y )
18	17	ineq2d 3634	. . . 4 |- ( x = y -> ( B i^i ~P x ) = ( B i^i ~P y ) )
19	16, 18	impbid1 207	. . 3 |- ( ( x e. ( topGen ` B ) /\ y e. ( topGen ` B ) ) -> ( ( B i^i ~P x ) = ( B i^i ~P y ) <-> x = y ) )
20	8, 19	dom2 7612	. 2 |- ( ~P B e. _V -> ( topGen ` B ) ~<_ ~P B )
21	1, 20	syl 17	1 |- ( B e. V -> ( topGen ` B ) ~<_ ~P B )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1444   e. wcel 1887   _Vcvv 3045   i^i cin 3403   C_ wss 3404   ~Pcpw 3951   U.cuni 4198   class class class wbr 4402   ` cfv 5582   ~<_ cdom 7567   topGenctg 15336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-dom 7571  df-topgen 15342

This theorem is referenced by:  2ndcredom  20465  kelac2lem  35922