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Theorem cnrsfin 24857
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 16590 . . . . 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 2256 . . . . 5  |-  U. J  =  U. J
76cnss1 16932 . . . 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 3123 . 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 3094   U.cuni 3768   ` cfv 4638  (class class class)co 5757   Topctop 16558  TopOnctopon 16559    Cn ccn 16881
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 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-map 6707  df-top 16563  df-topon 16566  df-cn 16884
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