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Theorem cnrsfin 25536
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 16665 . . . . 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 2285 . . . . 5  |-  U. J  =  U. J
76cnss1 17007 . . . 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 3183 . 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 1625    e. wcel 1686    C_ wss 3154   U.cuni 3829   ` cfv 5257  (class class class)co 5860   Topctop 16633  TopOnctopon 16634    Cn ccn 16956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-map 6776  df-top 16638  df-topon 16641  df-cn 16959
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