Theorem tgdom 20071

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 4585	. 2 |- ( B e. V -> ~P B e. _V )
2		inss1 3643	. . . . 5 |- ( B i^i ~P x ) C_ B
3		vex 3034	. . . . . . . 8 |- x e. _V
4	3	pwex 4584	. . . . . . 7 |- ~P x e. _V
5	4	inex2 4538	. . . . . 6 |- ( B i^i ~P x ) e. _V
6	5	elpw 3948	. . . . 5 |- ( ( B i^i ~P x ) e. ~P B <-> ( B i^i ~P x ) C_ B )
7	2, 6	mpbir 214	. . . 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 4198	. . . . . . 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 473	. . . . . 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 20052	. . . . . . 7 |- ( x e. ( topGen ` B ) -> x = U. ( B i^i ~P x ) )
12	11	ad2antrr 740	. . . . . 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 20052	. . . . . . 7 |- ( y e. ( topGen ` B ) -> y = U. ( B i^i ~P y ) )
14	13	ad2antlr 741	. . . . . 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 2515	. . . . 5 |- ( ( ( x e. ( topGen ` B ) /\ y e. ( topGen ` B ) ) /\ ( B i^i ~P x ) = ( B i^i ~P y ) ) -> x = y )
16	15	ex 441	. . . 4 |- ( ( x e. ( topGen ` B ) /\ y e. ( topGen ` B ) ) -> ( ( B i^i ~P x ) = ( B i^i ~P y ) -> x = y ) )
17		pweq 3945	. . . . 5 |- ( x = y -> ~P x = ~P y )
18	17	ineq2d 3625	. . . 4 |- ( x = y -> ( B i^i ~P x ) = ( B i^i ~P y ) )
19	16, 18	impbid1 208	. . 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 7630	. 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 376   = wceq 1452   e. wcel 1904   _Vcvv 3031   i^i cin 3389   C_ wss 3390   ~Pcpw 3942   U.cuni 4190   class class class wbr 4395   ` cfv 5589   ~<_ cdom 7585   topGenctg 15414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-dom 7589  df-topgen 15420

This theorem is referenced by:  2ndcredom  20542  kelac2lem  35993