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Theorem cnrsfin 24691
Description: A mapping remains continuous when the topology associated to its domain is replaced by a finer one. (Contributed by FL, 22-May-2008.)
Assertion
Ref Expression
cnrsfin  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  ->  (
( F  e.  ( J  Cn  L )  /\  U. J  = 
U. K  /\  J  C_  K )  ->  F  e.  ( K  Cn  L
) ) )

Proof of Theorem cnrsfin
StepHypRef Expression
1 simpl2 964 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  /\  ( F  e.  ( J  Cn  L )  /\  U. J  = 
U. K  /\  J  C_  K ) )  ->  K  e.  Top )
2 simpr2 967 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  /\  ( F  e.  ( J  Cn  L )  /\  U. J  = 
U. K  /\  J  C_  K ) )  ->  U. J  =  U. K )
3 istopon 16495 . . . . 5  |-  ( K  e.  (TopOn `  U. J )  <->  ( K  e.  Top  /\  U. J  =  U. K ) )
41, 2, 3sylanbrc 648 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  /\  ( F  e.  ( J  Cn  L )  /\  U. J  = 
U. K  /\  J  C_  K ) )  ->  K  e.  (TopOn `  U. J ) )
5 simpr3 968 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  /\  ( F  e.  ( J  Cn  L )  /\  U. J  = 
U. K  /\  J  C_  K ) )  ->  J  C_  K )
6 eqid 2253 . . . . 5  |-  U. J  =  U. J
76cnss1 16837 . . . 4  |-  ( ( K  e.  (TopOn `  U. J )  /\  J  C_  K )  ->  ( J  Cn  L )  C_  ( K  Cn  L
) )
84, 5, 7syl2anc 645 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  /\  ( F  e.  ( J  Cn  L )  /\  U. J  = 
U. K  /\  J  C_  K ) )  -> 
( J  Cn  L
)  C_  ( K  Cn  L ) )
9 simpr1 966 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  /\  ( F  e.  ( J  Cn  L )  /\  U. J  = 
U. K  /\  J  C_  K ) )  ->  F  e.  ( J  Cn  L ) )
108, 9sseldd 3104 . 2  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  /\  ( F  e.  ( J  Cn  L )  /\  U. J  = 
U. K  /\  J  C_  K ) )  ->  F  e.  ( K  Cn  L ) )
1110ex 425 1  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  ->  (
( F  e.  ( J  Cn  L )  /\  U. J  = 
U. K  /\  J  C_  K )  ->  F  e.  ( K  Cn  L
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    C_ wss 3078   U.cuni 3727   ` cfv 4592  (class class class)co 5710   Topctop 16463  TopOnctopon 16464    Cn ccn 16786
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-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-fn 4603  df-f 4604  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-map 6660  df-top 16468  df-topon 16471  df-cn 16789
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