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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.
StepHypRef 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
43pwex 4584 . . . . . . 7  |-  ~P x  e.  _V
54inex2 4538 . . . . . 6  |-  ( B  i^i  ~P x )  e.  _V
65elpw 3948 . . . . 5  |-  ( ( B  i^i  ~P x
)  e.  ~P B  <->  ( B  i^i  ~P x
)  C_  B )
72, 6mpbir 214 . . . 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 4198 . . . . . . 7  |-  ( ( B  i^i  ~P x
)  =  ( B  i^i  ~P y )  ->  U. ( B  i^i  ~P x )  =  U. ( B  i^i  ~P y
) )
109adantl 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 ) )
1211ad2antrr 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 ) )
1413ad2antlr 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 ) )
1510, 12, 143eqtr4d 2515 . . . . 5  |-  ( ( ( x  e.  (
topGen `  B )  /\  y  e.  ( topGen `  B ) )  /\  ( B  i^i  ~P x
)  =  ( B  i^i  ~P y ) )  ->  x  =  y )
1615ex 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
)
1817ineq2d 3625 . . . 4  |-  ( x  =  y  ->  ( B  i^i  ~P x )  =  ( B  i^i  ~P y ) )
1916, 18impbid1 208 . . 3  |-  ( ( x  e.  ( topGen `  B )  /\  y  e.  ( topGen `  B )
)  ->  ( ( B  i^i  ~P x )  =  ( B  i^i  ~P y )  <->  x  =  y ) )
208, 19dom2 7630 . 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 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
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