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Theorem cnss1 20369
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 2471 . . . . . 6  |-  U. L  =  U. L
31, 2cnf 20339 . . . . 5  |-  ( f  e.  ( J  Cn  L )  ->  f : X --> U. L )
43adantl 473 . . . 4  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L
) )  ->  f : X --> U. L )
5 simpllr 777 . . . . . 6  |-  ( ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L ) )  /\  x  e.  L )  ->  J  C_  K )
6 cnima 20358 . . . . . . 7  |-  ( ( f  e.  ( J  Cn  L )  /\  x  e.  L )  ->  ( `' f "
x )  e.  J
)
76adantll 728 . . . . . 6  |-  ( ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L ) )  /\  x  e.  L )  ->  ( `' f " x
)  e.  J )
85, 7sseldd 3419 . . . . 5  |-  ( ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L ) )  /\  x  e.  L )  ->  ( `' f " x
)  e.  K )
98ralrimiva 2809 . . . 4  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L
) )  ->  A. x  e.  L  ( `' f " x )  e.  K )
10 simpll 768 . . . . 5  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L
) )  ->  K  e.  (TopOn `  X )
)
11 cntop2 20334 . . . . . . 7  |-  ( f  e.  ( J  Cn  L )  ->  L  e.  Top )
1211adantl 473 . . . . . 6  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L
) )  ->  L  e.  Top )
132toptopon 20025 . . . . . 6  |-  ( L  e.  Top  <->  L  e.  (TopOn `  U. L ) )
1412, 13sylib 201 . . . . 5  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L
) )  ->  L  e.  (TopOn `  U. L ) )
15 iscn 20328 . . . . 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 673 . . . 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 936 . . 3  |-  ( ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  /\  f  e.  ( J  Cn  L
) )  ->  f  e.  ( K  Cn  L
) )
1817ex 441 . 2  |-  ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  (
f  e.  ( J  Cn  L )  -> 
f  e.  ( K  Cn  L ) ) )
1918ssrdv 3424 1  |-  ( ( K  e.  (TopOn `  X )  /\  J  C_  K )  ->  ( J  Cn  L )  C_  ( K  Cn  L
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756    C_ wss 3390   U.cuni 4190   `'ccnv 4838   "cima 4842   -->wf 5585   ` cfv 5589  (class class class)co 6308   Topctop 19994  TopOnctopon 19995    Cn ccn 20317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-map 7492  df-top 19998  df-topon 20000  df-cn 20320
This theorem is referenced by:  kgen2cn  20651  xkopjcn  20748
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