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Theorem cnss1 17005
Description: If the topology  K is finer than  J, then there are more continuous functions from  K than from  J. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
Hypothesis
Ref Expression
cnss1.1  |-  X  = 
U. J
Assertion
Ref Expression
cnss1  |-  ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( J  Cn  L )  C_  ( K  Cn  L
) )

Proof of Theorem cnss1
Dummy variables  x  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnss1.1 . . . . . 6  |-  X  = 
U. J
2 eqid 2283 . . . . . 6  |-  U. L  =  U. L
31, 2cnf 16976 . . . . 5  |-  ( f  e.  ( J  Cn  L )  ->  f : X --> U. L )
43adantl 452 . . . 4  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L
) )  ->  f : X --> U. L )
5 simpllr 735 . . . . . 6  |-  ( ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L ) )  /\  x  e.  L )  ->  J  C_  K )
6 cnima 16994 . . . . . . 7  |-  ( ( f  e.  ( J  Cn  L )  /\  x  e.  L )  ->  ( `' f "
x )  e.  J
)
76adantll 694 . . . . . 6  |-  ( ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L ) )  /\  x  e.  L )  ->  ( `' f " x
)  e.  J )
85, 7sseldd 3181 . . . . 5  |-  ( ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L ) )  /\  x  e.  L )  ->  ( `' f " x
)  e.  K )
98ralrimiva 2626 . . . 4  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L
) )  ->  A. x  e.  L  ( `' f " x )  e.  K )
10 simpll 730 . . . . 5  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L
) )  ->  K  e.  (TopOn `  X )
)
11 cntop2 16971 . . . . . . 7  |-  ( f  e.  ( J  Cn  L )  ->  L  e.  Top )
1211adantl 452 . . . . . 6  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L
) )  ->  L  e.  Top )
132toptopon 16671 . . . . . 6  |-  ( L  e.  Top  <->  L  e.  (TopOn `  U. L ) )
1412, 13sylib 188 . . . . 5  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L
) )  ->  L  e.  (TopOn `  U. L ) )
15 iscn 16965 . . . . 5  |-  ( ( K  e.  (TopOn `  X )  /\  L  e.  (TopOn `  U. L ) )  ->  ( f  e.  ( K  Cn  L
)  <->  ( f : X --> U. L  /\  A. x  e.  L  ( `' f " x
)  e.  K ) ) )
1610, 14, 15syl2anc 642 . . . 4  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L
) )  ->  (
f  e.  ( K  Cn  L )  <->  ( f : X --> U. L  /\  A. x  e.  L  ( `' f " x
)  e.  K ) ) )
174, 9, 16mpbir2and 888 . . 3  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L
) )  ->  f  e.  ( K  Cn  L
) )
1817ex 423 . 2  |-  ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  (
f  e.  ( J  Cn  L )  -> 
f  e.  ( K  Cn  L ) ) )
1918ssrdv 3185 1  |-  ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( J  Cn  L )  C_  ( K  Cn  L
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   U.cuni 3827   `'ccnv 4688   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   Topctop 16631  TopOnctopon 16632    Cn ccn 16954
This theorem is referenced by:  kgen2cn  17254  xkopjcn  17350  cnrsfin  25525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957
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