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Theorem cnss2 17263
Description: If the topology  K is finer than  J, then there are fewer continuous functions into  K than into  J from some other space. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
Hypothesis
Ref Expression
cnss2.1  |-  Y  = 
U. K
Assertion
Ref Expression
cnss2  |-  ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  ->  ( J  Cn  K )  C_  ( J  Cn  L
) )

Proof of Theorem cnss2
Dummy variables  x  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2387 . . . . . 6  |-  U. J  =  U. J
2 cnss2.1 . . . . . 6  |-  Y  = 
U. K
31, 2cnf 17232 . . . . 5  |-  ( f  e.  ( J  Cn  K )  ->  f : U. J --> Y )
43adantl 453 . . . 4  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  f : U. J --> Y )
5 simplr 732 . . . . 5  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  L  C_  K )
6 cnima 17251 . . . . . . 7  |-  ( ( f  e.  ( J  Cn  K )  /\  x  e.  K )  ->  ( `' f "
x )  e.  J
)
76ralrimiva 2732 . . . . . 6  |-  ( f  e.  ( J  Cn  K )  ->  A. x  e.  K  ( `' f " x )  e.  J )
87adantl 453 . . . . 5  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  A. x  e.  K  ( `' f " x )  e.  J )
9 ssralv 3350 . . . . 5  |-  ( L 
C_  K  ->  ( A. x  e.  K  ( `' f " x
)  e.  J  ->  A. x  e.  L  ( `' f " x
)  e.  J ) )
105, 8, 9sylc 58 . . . 4  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  A. x  e.  L  ( `' f " x )  e.  J )
11 cntop1 17226 . . . . . . 7  |-  ( f  e.  ( J  Cn  K )  ->  J  e.  Top )
1211adantl 453 . . . . . 6  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  J  e.  Top )
131toptopon 16921 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
1412, 13sylib 189 . . . . 5  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  J  e.  (TopOn `  U. J ) )
15 simpll 731 . . . . 5  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  L  e.  (TopOn `  Y )
)
16 iscn 17221 . . . . 5  |-  ( ( J  e.  (TopOn `  U. J )  /\  L  e.  (TopOn `  Y )
)  ->  ( f  e.  ( J  Cn  L
)  <->  ( f : U. J --> Y  /\  A. x  e.  L  ( `' f " x
)  e.  J ) ) )
1714, 15, 16syl2anc 643 . . . 4  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  (
f  e.  ( J  Cn  L )  <->  ( f : U. J --> Y  /\  A. x  e.  L  ( `' f " x
)  e.  J ) ) )
184, 10, 17mpbir2and 889 . . 3  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  f  e.  ( J  Cn  L
) )
1918ex 424 . 2  |-  ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  ->  (
f  e.  ( J  Cn  K )  -> 
f  e.  ( J  Cn  L ) ) )
2019ssrdv 3297 1  |-  ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  ->  ( J  Cn  K )  C_  ( J  Cn  L
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649    C_ wss 3263   U.cuni 3957   `'ccnv 4817   "cima 4821   -->wf 5390   ` cfv 5394  (class class class)co 6020   Topctop 16881  TopOnctopon 16882    Cn ccn 17210
This theorem is referenced by:  kgencn3  17511  xmetdcn  18740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-map 6956  df-top 16886  df-topon 16889  df-cn 17213
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