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Theorem cnres2 26370
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 985 . . 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 983 . . 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 17311 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
51, 2, 4syl2anc 643 . 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 982 . . . 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 16961 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
96, 8sylib 189 . . 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 4858 . . . 4  |-  ( F
" A )  =  ran  ( F  |`  A )
11 simp3r 986 . . . . 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 17272 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
13 ffun 5560 . . . . . . 7  |-  ( F : X --> Y  ->  Fun  F )
141, 12, 133syl 19 . . . . . 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 5562 . . . . . . . 8  |-  ( F : X --> Y  ->  dom  F  =  X )
161, 12, 153syl 19 . . . . . . 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 3353 . . . . . 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 5744 . . . . . 6  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A. x  e.  A  ( F `  x )  e.  B ) )
1914, 17, 18syl2anc 643 . . . . 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 224 . . . 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 3361 . . 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 984 . . 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 17312 . . 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 1184 . 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 202 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2674    C_ wss 3288   U.cuni 3983   dom cdm 4845   ran crn 4846    |` cres 4847   "cima 4848   Fun wfun 5415   -->wf 5417   ` cfv 5421  (class class class)co 6048   ↾t crest 13611   Topctop 16921  TopOnctopon 16922    Cn ccn 17250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-fin 7080  df-fi 7382  df-rest 13613  df-topgen 13630  df-top 16926  df-bases 16928  df-topon 16929  df-cn 17253
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