Theorem tgdom 14240

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 4209	. 2 |- ( B e. V -> ~P B e. _V )
2		inss1 3379	. . . . 5 |- ( B i^i ~P x ) C_ B
3		vpwex 4208	. . . . . . 7 |- ~P x e. _V
4	3	inex2 4164	. . . . . 6 |- ( B i^i ~P x ) e. _V
5	4	elpw 3607	. . . . 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 3844	. . . . . . 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 14223	. . . . . . 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 14223	. . . . . . 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 2236	. . . . 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 3604	. . . . 5 |- ( x = y -> ~P x = ~P y )
17	16	ineq2d 3360	. . . 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 6829	. 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 1364   e. wcel 2164   _Vcvv 2760   i^i cin 3152   C_ wss 3153   ~Pcpw 3601   U.cuni 3835   class class class wbr 4029   ` cfv 5254   ~<_ cdom 6793   topGenctg 12865

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-dom 6796  df-topgen 12871

This theorem is referenced by: (None)