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Theorem topmtcl 25478
Description: The meet of a collection of topologies on  X is again a topology on  X. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
topmtcl  |-  ( ( X  e.  V  /\  S  C_  (TopOn `  X
) )  ->  ( ~P X  i^i  |^| S
)  e.  (TopOn `  X ) )

Proof of Theorem topmtcl
StepHypRef Expression
1 toponmre 16662 . 2  |-  ( X  e.  V  ->  (TopOn `  X )  e.  (Moore `  ~P X ) )
2 mrerintcl 13371 . 2  |-  ( ( (TopOn `  X )  e.  (Moore `  ~P X )  /\  S  C_  (TopOn `  X ) )  -> 
( ~P X  i^i  |^| S )  e.  (TopOn `  X ) )
31, 2sylan 459 1  |-  ( ( X  e.  V  /\  S  C_  (TopOn `  X
) )  ->  ( ~P X  i^i  |^| S
)  e.  (TopOn `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1621    i^i cin 3077    C_ wss 3078   ~Pcpw 3530   |^|cint 3760   ` cfv 4592  Moorecmre 13358  TopOnctopon 16464
This theorem is referenced by:  topmeet  25479
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-mre 13361  df-top 16468  df-topon 16471
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