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Theorem cnrsfin 24893
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 16626 . . . . 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 2258 . . . . 5  |-  U. J  =  U. J
76cnss1 16968 . . . 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 3156 . 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 3127   U.cuni 3801   ` cfv 4673  (class class class)co 5792   Topctop 16594  TopOnctopon 16595    Cn ccn 16917
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 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-map 6742  df-top 16599  df-topon 16602  df-cn 16920
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