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Theorem cnrsfin 25628
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 959 . . . . 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 962 . . . . 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 16679 . . . . 5  |-  ( K  e.  (TopOn `  U. J )  <->  ( K  e.  Top  /\  U. J  =  U. K ) )
41, 2, 3sylanbrc 645 . . . 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 963 . . . 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 2296 . . . . 5  |-  U. J  =  U. J
76cnss1 17021 . . . 4  |-  ( ( K  e.  (TopOn `  U. J )  /\  J  C_  K )  ->  ( J  Cn  L )  C_  ( K  Cn  L
) )
84, 5, 7syl2anc 642 . . 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 961 . . 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 3194 . 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 423 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 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   U.cuni 3843   ` cfv 5271  (class class class)co 5874   Topctop 16647  TopOnctopon 16648    Cn ccn 16970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-top 16652  df-topon 16655  df-cn 16973
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