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Theorem cnss2 17022
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 2296 . . . . . 6  |-  U. J  =  U. J
2 cnss2.1 . . . . . 6  |-  Y  = 
U. K
31, 2cnf 16992 . . . . 5  |-  ( f  e.  ( J  Cn  K )  ->  f : U. J --> Y )
43adantl 452 . . . 4  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  f : U. J --> Y )
5 simplr 731 . . . . 5  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  L  C_  K )
6 cnima 17010 . . . . . . 7  |-  ( ( f  e.  ( J  Cn  K )  /\  x  e.  K )  ->  ( `' f "
x )  e.  J
)
76ralrimiva 2639 . . . . . 6  |-  ( f  e.  ( J  Cn  K )  ->  A. x  e.  K  ( `' f " x )  e.  J )
87adantl 452 . . . . 5  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  A. x  e.  K  ( `' f " x )  e.  J )
9 ssralv 3250 . . . . 5  |-  ( L 
C_  K  ->  ( A. x  e.  K  ( `' f " x
)  e.  J  ->  A. x  e.  L  ( `' f " x
)  e.  J ) )
105, 8, 9sylc 56 . . . 4  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  A. x  e.  L  ( `' f " x )  e.  J )
11 cntop1 16986 . . . . . . 7  |-  ( f  e.  ( J  Cn  K )  ->  J  e.  Top )
1211adantl 452 . . . . . 6  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  J  e.  Top )
131toptopon 16687 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
1412, 13sylib 188 . . . . 5  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  J  e.  (TopOn `  U. J ) )
15 simpll 730 . . . . 5  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  L  e.  (TopOn `  Y )
)
16 iscn 16981 . . . . 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 642 . . . 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 888 . . 3  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  f  e.  ( J  Cn  L
) )
1918ex 423 . 2  |-  ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  ->  (
f  e.  ( J  Cn  K )  -> 
f  e.  ( J  Cn  L ) ) )
2019ssrdv 3198 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   U.cuni 3843   `'ccnv 4704   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874   Topctop 16647  TopOnctopon 16648    Cn ccn 16970
This theorem is referenced by:  kgencn3  17269  xmetdcn  18359  cnrscoa  25629
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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