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Theorem cnres2 32088
Description: The restriction of a continuous function to a subset is continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
Hypotheses
Ref Expression
cnres2.1  |-  X  = 
U. J
cnres2.2  |-  Y  = 
U. K
Assertion
Ref Expression
cnres2  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  ( Kt  B ) ) )
Distinct variable groups:    x, J    x, K    x, F    x, X    x, Y    x, A    x, B

Proof of Theorem cnres2
StepHypRef Expression
1 simp3l 1035 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  F  e.  ( J  Cn  K
) )
2 simp2l 1033 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  A  C_  X )
3 cnres2.1 . . . 4  |-  X  = 
U. J
43cnrest 20294 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
51, 2, 4syl2anc 666 . 2  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
6 simp1r 1032 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  K  e.  Top )
7 cnres2.2 . . . . 5  |-  Y  = 
U. K
87toptopon 19941 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
96, 8sylib 200 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  K  e.  (TopOn `  Y )
)
10 df-ima 4846 . . . 4  |-  ( F
" A )  =  ran  ( F  |`  A )
11 simp3r 1036 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  A. x  e.  A  ( F `  x )  e.  B
)
123, 7cnf 20255 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
13 ffun 5729 . . . . . . 7  |-  ( F : X --> Y  ->  Fun  F )
141, 12, 133syl 18 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  Fun  F )
15 fdm 5731 . . . . . . . 8  |-  ( F : X --> Y  ->  dom  F  =  X )
161, 12, 153syl 18 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  dom  F  =  X )
172, 16sseqtr4d 3468 . . . . . 6  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  A  C_ 
dom  F )
18 funimass4 5914 . . . . . 6  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A. x  e.  A  ( F `  x )  e.  B ) )
1914, 17, 18syl2anc 666 . . . . 5  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  (
( F " A
)  C_  B  <->  A. x  e.  A  ( F `  x )  e.  B
) )
2011, 19mpbird 236 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  ( F " A )  C_  B )
2110, 20syl5eqssr 3476 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  ran  ( F  |`  A ) 
C_  B )
22 simp2r 1034 . . 3  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  B  C_  Y )
23 cnrest2 20295 . . 3  |-  ( ( K  e.  (TopOn `  Y )  /\  ran  ( F  |`  A ) 
C_  B  /\  B  C_  Y )  ->  (
( F  |`  A )  e.  ( ( Jt  A )  Cn  K )  <-> 
( F  |`  A )  e.  ( ( Jt  A )  Cn  ( Kt  B ) ) ) )
249, 21, 22, 23syl3anc 1267 . 2  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  (
( F  |`  A )  e.  ( ( Jt  A )  Cn  K )  <-> 
( F  |`  A )  e.  ( ( Jt  A )  Cn  ( Kt  B ) ) ) )
255, 24mpbid 214 1  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( A  C_  X  /\  B  C_  Y )  /\  ( F  e.  ( J  Cn  K
)  /\  A. x  e.  A  ( F `  x )  e.  B
) )  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  ( Kt  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   A.wral 2736    C_ wss 3403   U.cuni 4197   dom cdm 4833   ran crn 4834    |` cres 4835   "cima 4836   Fun wfun 5575   -->wf 5577   ` cfv 5581  (class class class)co 6288   ↾t crest 15312   Topctop 19910  TopOnctopon 19911    Cn ccn 20233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-oadd 7183  df-er 7360  df-map 7471  df-en 7567  df-fin 7570  df-fi 7922  df-rest 15314  df-topgen 15335  df-top 19914  df-bases 19915  df-topon 19916  df-cn 20236
This theorem is referenced by: (None)
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