Theorem tgdom 13657

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 4182	. 2 |- ( B e. V -> ~P B e. _V )
2		inss1 3357	. . . . 5 |- ( B i^i ~P x ) C_ B
3		vpwex 4181	. . . . . . 7 |- ~P x e. _V
4	3	inex2 4140	. . . . . 6 |- ( B i^i ~P x ) e. _V
5	4	elpw 3583	. . . . 5 |- ( ( B i^i ~P x ) e. ~P B <-> ( B i^i ~P x ) C_ B )
6	2, 5	mpbir 146	. . . 4 |- ( B i^i ~P x ) e. ~P B
7	6	a1i 9	. . 3 |- ( x e. ( topGen ` B ) -> ( B i^i ~P x ) e. ~P B )
8		unieq 3820	. . . . . . 7 |- ( ( B i^i ~P x ) = ( B i^i ~P y ) -> U. ( B i^i ~P x ) = U. ( B i^i ~P y ) )
9	8	adantl 277	. . . . . 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 ) )
10		eltg4i 13640	. . . . . . 7 |- ( x e. ( topGen ` B ) -> x = U. ( B i^i ~P x ) )
11	10	ad2antrr 488	. . . . . 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 ) )
12		eltg4i 13640	. . . . . . 7 |- ( y e. ( topGen ` B ) -> y = U. ( B i^i ~P y ) )
13	12	ad2antlr 489	. . . . . 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 ) )
14	9, 11, 13	3eqtr4d 2220	. . . . 5 |- ( ( ( x e. ( topGen ` B ) /\ y e. ( topGen ` B ) ) /\ ( B i^i ~P x ) = ( B i^i ~P y ) ) -> x = y )
15	14	ex 115	. . . 4 |- ( ( x e. ( topGen ` B ) /\ y e. ( topGen ` B ) ) -> ( ( B i^i ~P x ) = ( B i^i ~P y ) -> x = y ) )
16		pweq 3580	. . . . 5 |- ( x = y -> ~P x = ~P y )
17	16	ineq2d 3338	. . . 4 |- ( x = y -> ( B i^i ~P x ) = ( B i^i ~P y ) )
18	15, 17	impbid1 142	. . 3 |- ( ( x e. ( topGen ` B ) /\ y e. ( topGen ` B ) ) -> ( ( B i^i ~P x ) = ( B i^i ~P y ) <-> x = y ) )
19	7, 18	dom2 6777	. 2 |- ( ~P B e. _V -> ( topGen ` B ) ~<_ ~P B )
20	1, 19	syl 14	1 |- ( B e. V -> ( topGen ` B ) ~<_ ~P B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2739   i^i cin 3130   C_ wss 3131   ~Pcpw 3577   U.cuni 3811   class class class wbr 4005   ` cfv 5218   ~<_ cdom 6741   topGenctg 12708

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-dom 6744  df-topgen 12714

This theorem is referenced by: (None)