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Theorem tgdom 16716
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 4194 . 2  |-  ( B  e.  V  ->  ~P B  e.  _V )
2 inss1 3389 . . . . 5  |-  ( B  i^i  ~P x ) 
C_  B
3 vex 2791 . . . . . . . 8  |-  x  e. 
_V
43pwex 4193 . . . . . . 7  |-  ~P x  e.  _V
54inex2 4156 . . . . . 6  |-  ( B  i^i  ~P x )  e.  _V
65elpw 3631 . . . . 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 3836 . . . . . . 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 16698 . . . . . . 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 16698 . . . . . . 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 2325 . . . . 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 3628 . . . . 5  |-  ( x  =  y  ->  ~P x  =  ~P y
)
1817ineq2d 3370 . . . 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 6904 . 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 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   class class class wbr 4023   ` cfv 5255    ~<_ cdom 6861   topGenctg 13342
This theorem is referenced by:  2ndcredom  17176  kelac2lem  26574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-dom 6865  df-topgen 13344
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