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Theorem tgdom 16732
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.
StepHypRef Expression
1 pwexg 4210 . 2  |-  ( B  e.  V  ->  ~P B  e.  _V )
2 inss1 3402 . . . . 5  |-  ( B  i^i  ~P x ) 
C_  B
3 vex 2804 . . . . . . . 8  |-  x  e. 
_V
43pwex 4209 . . . . . . 7  |-  ~P x  e.  _V
54inex2 4172 . . . . . 6  |-  ( B  i^i  ~P x )  e.  _V
65elpw 3644 . . . . 5  |-  ( ( B  i^i  ~P x
)  e.  ~P B  <->  ( B  i^i  ~P x
)  C_  B )
72, 6mpbir 200 . . . 4  |-  ( B  i^i  ~P x )  e.  ~P B
87a1i 10 . . 3  |-  ( x  e.  ( topGen `  B
)  ->  ( B  i^i  ~P x )  e. 
~P B )
9 unieq 3852 . . . . . . 7  |-  ( ( B  i^i  ~P x
)  =  ( B  i^i  ~P y )  ->  U. ( B  i^i  ~P x )  =  U. ( B  i^i  ~P y
) )
109adantl 452 . . . . . 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 16714 . . . . . . 7  |-  ( x  e.  ( topGen `  B
)  ->  x  =  U. ( B  i^i  ~P x ) )
1211ad2antrr 706 . . . . . 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 16714 . . . . . . 7  |-  ( y  e.  ( topGen `  B
)  ->  y  =  U. ( B  i^i  ~P y ) )
1413ad2antlr 707 . . . . . 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 ) )
1510, 12, 143eqtr4d 2338 . . . . 5  |-  ( ( ( x  e.  (
topGen `  B )  /\  y  e.  ( topGen `  B ) )  /\  ( B  i^i  ~P x
)  =  ( B  i^i  ~P y ) )  ->  x  =  y )
1615ex 423 . . . 4  |-  ( ( x  e.  ( topGen `  B )  /\  y  e.  ( topGen `  B )
)  ->  ( ( B  i^i  ~P x )  =  ( B  i^i  ~P y )  ->  x  =  y ) )
17 pweq 3641 . . . . 5  |-  ( x  =  y  ->  ~P x  =  ~P y
)
1817ineq2d 3383 . . . 4  |-  ( x  =  y  ->  ( B  i^i  ~P x )  =  ( B  i^i  ~P y ) )
1916, 18impbid1 194 . . 3  |-  ( ( x  e.  ( topGen `  B )  /\  y  e.  ( topGen `  B )
)  ->  ( ( B  i^i  ~P x )  =  ( B  i^i  ~P y )  <->  x  =  y ) )
208, 19dom2 6920 . 2  |-  ( ~P B  e.  _V  ->  (
topGen `  B )  ~<_  ~P B )
211, 20syl 15 1  |-  ( B  e.  V  ->  ( topGen `
 B )  ~<_  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   class class class wbr 4039   ` cfv 5271    ~<_ cdom 6877   topGenctg 13358
This theorem is referenced by:  2ndcredom  17192  kelac2lem  27265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-dom 6881  df-topgen 13360
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