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Theorem cnss2 12406
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 2139 . . . . . 6  |-  U. J  =  U. J
2 cnss2.1 . . . . . 6  |-  Y  = 
U. K
31, 2cnf 12383 . . . . 5  |-  ( f  e.  ( J  Cn  K )  ->  f : U. J --> Y )
43adantl 275 . . . 4  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  f : U. J --> Y )
5 simplr 519 . . . . 5  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  L  C_  K )
6 cnima 12399 . . . . . . 7  |-  ( ( f  e.  ( J  Cn  K )  /\  x  e.  K )  ->  ( `' f "
x )  e.  J
)
76ralrimiva 2505 . . . . . 6  |-  ( f  e.  ( J  Cn  K )  ->  A. x  e.  K  ( `' f " x )  e.  J )
87adantl 275 . . . . 5  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  A. x  e.  K  ( `' f " x )  e.  J )
9 ssralv 3161 . . . . 5  |-  ( L 
C_  K  ->  ( A. x  e.  K  ( `' f " x
)  e.  J  ->  A. x  e.  L  ( `' f " x
)  e.  J ) )
105, 8, 9sylc 62 . . . 4  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  A. x  e.  L  ( `' f " x )  e.  J )
11 cntop1 12380 . . . . . . 7  |-  ( f  e.  ( J  Cn  K )  ->  J  e.  Top )
1211adantl 275 . . . . . 6  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  J  e.  Top )
131toptopon 12195 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
1412, 13sylib 121 . . . . 5  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  J  e.  (TopOn `  U. J ) )
15 simpll 518 . . . . 5  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  L  e.  (TopOn `  Y )
)
16 iscn 12376 . . . . 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 408 . . . 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 928 . . 3  |-  ( ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  /\  f  e.  ( J  Cn  K
) )  ->  f  e.  ( J  Cn  L
) )
1918ex 114 . 2  |-  ( ( L  e.  (TopOn `  Y )  /\  L  C_  K )  ->  (
f  e.  ( J  Cn  K )  -> 
f  e.  ( J  Cn  L ) ) )
2019ssrdv 3103 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    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   A.wral 2416    C_ wss 3071   U.cuni 3736   `'ccnv 4538   "cima 4542   -->wf 5119   ` cfv 5123  (class class class)co 5774   Topctop 12174  TopOnctopon 12187    Cn ccn 12364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-top 12175  df-topon 12188  df-cn 12367
This theorem is referenced by: (None)
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