Theorem tgdom 18605

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 4497	. 2 |- ( B e. V -> ~P B e. _V )
2		inss1 3591	. . . . 5 |- ( B i^i ~P x ) C_ B
3		vex 2996	. . . . . . . 8 |- x e. _V
4	3	pwex 4496	. . . . . . 7 |- ~P x e. _V
5	4	inex2 4455	. . . . . 6 |- ( B i^i ~P x ) e. _V
6	5	elpw 3887	. . . . 5 |- ( ( B i^i ~P x ) e. ~P B <-> ( B i^i ~P x ) C_ B )
7	2, 6	mpbir 209	. . . 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 4120	. . . . . . 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 466	. . . . . 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 18587	. . . . . . 7 |- ( x e. ( topGen ` B ) -> x = U. ( B i^i ~P x ) )
12	11	ad2antrr 725	. . . . . 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 18587	. . . . . . 7 |- ( y e. ( topGen ` B ) -> y = U. ( B i^i ~P y ) )
14	13	ad2antlr 726	. . . . . 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 2485	. . . . 5 |- ( ( ( x e. ( topGen ` B ) /\ y e. ( topGen ` B ) ) /\ ( B i^i ~P x ) = ( B i^i ~P y ) ) -> x = y )
16	15	ex 434	. . . 4 |- ( ( x e. ( topGen ` B ) /\ y e. ( topGen ` B ) ) -> ( ( B i^i ~P x ) = ( B i^i ~P y ) -> x = y ) )
17		pweq 3884	. . . . 5 |- ( x = y -> ~P x = ~P y )
18	17	ineq2d 3573	. . . 4 |- ( x = y -> ( B i^i ~P x ) = ( B i^i ~P y ) )
19	16, 18	impbid1 203	. . 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 7373	. 2 |- ( ~P B e. _V -> ( topGen ` B ) ~<_ ~P B )
21	1, 20	syl 16	1 |- ( B e. V -> ( topGen ` B ) ~<_ ~P B )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   _Vcvv 2993   i^i cin 3348   C_ wss 3349   ~Pcpw 3881   U.cuni 4112   class class class wbr 4313   ` cfv 5439   ~<_ cdom 7329   topGenctg 14397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-dom 7333  df-topgen 14403

This theorem is referenced by:  2ndcredom  19076  kelac2lem  29443