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Theorem tgdom 19925
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 4609 . 2  |-  ( B  e.  V  ->  ~P B  e.  _V )
2 inss1 3688 . . . . 5  |-  ( B  i^i  ~P x ) 
C_  B
3 vex 3090 . . . . . . . 8  |-  x  e. 
_V
43pwex 4608 . . . . . . 7  |-  ~P x  e.  _V
54inex2 4567 . . . . . 6  |-  ( B  i^i  ~P x )  e.  _V
65elpw 3991 . . . . 5  |-  ( ( B  i^i  ~P x
)  e.  ~P B  <->  ( B  i^i  ~P x
)  C_  B )
72, 6mpbir 212 . . . 4  |-  ( B  i^i  ~P x )  e.  ~P B
87a1i 11 . . 3  |-  ( x  e.  ( topGen `  B
)  ->  ( B  i^i  ~P x )  e. 
~P B )
9 unieq 4230 . . . . . . 7  |-  ( ( B  i^i  ~P x
)  =  ( B  i^i  ~P y )  ->  U. ( B  i^i  ~P x )  =  U. ( B  i^i  ~P y
) )
109adantl 467 . . . . . 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 19906 . . . . . . 7  |-  ( x  e.  ( topGen `  B
)  ->  x  =  U. ( B  i^i  ~P x ) )
1211ad2antrr 730 . . . . . 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 19906 . . . . . . 7  |-  ( y  e.  ( topGen `  B
)  ->  y  =  U. ( B  i^i  ~P y ) )
1413ad2antlr 731 . . . . . 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 2480 . . . . 5  |-  ( ( ( x  e.  (
topGen `  B )  /\  y  e.  ( topGen `  B ) )  /\  ( B  i^i  ~P x
)  =  ( B  i^i  ~P y ) )  ->  x  =  y )
1615ex 435 . . . 4  |-  ( ( x  e.  ( topGen `  B )  /\  y  e.  ( topGen `  B )
)  ->  ( ( B  i^i  ~P x )  =  ( B  i^i  ~P y )  ->  x  =  y ) )
17 pweq 3988 . . . . 5  |-  ( x  =  y  ->  ~P x  =  ~P y
)
1817ineq2d 3670 . . . 4  |-  ( x  =  y  ->  ( B  i^i  ~P x )  =  ( B  i^i  ~P y ) )
1916, 18impbid1 206 . . 3  |-  ( ( x  e.  ( topGen `  B )  /\  y  e.  ( topGen `  B )
)  ->  ( ( B  i^i  ~P x )  =  ( B  i^i  ~P y )  <->  x  =  y ) )
208, 19dom2 7619 . 2  |-  ( ~P B  e.  _V  ->  (
topGen `  B )  ~<_  ~P B )
211, 20syl 17 1  |-  ( B  e.  V  ->  ( topGen `
 B )  ~<_  ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087    i^i cin 3441    C_ wss 3442   ~Pcpw 3985   U.cuni 4222   class class class wbr 4426   ` cfv 5601    ~<_ cdom 7575   topGenctg 15295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-dom 7579  df-topgen 15301
This theorem is referenced by:  2ndcredom  20396  kelac2lem  35628
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