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Theorem cnres2 25989
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 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
) )  ->  F  e.  ( J  Cn  K
) )
2 simp2l 982 . . 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 17230 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
51, 2, 4syl2anc 642 . 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 981 . . . 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 16888 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
96, 8sylib 188 . . 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 4805 . . . 4  |-  ( F
" A )  =  ran  ( F  |`  A )
11 simp3r 985 . . . . 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 17193 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
13 ffun 5497 . . . . . . 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 5499 . . . . . . . 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 3301 . . . . . 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 5680 . . . . . 6  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A. x  e.  A  ( F `  x )  e.  B ) )
1914, 17, 18syl2anc 642 . . . . 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 223 . . . 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 3309 . . 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 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
) )  ->  B  C_  Y )
23 cnrest2 17231 . . 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 1183 . 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 201 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 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   A.wral 2628    C_ wss 3238   U.cuni 3929   dom cdm 4792   ran crn 4793    |` cres 4794   "cima 4795   Fun wfun 5352   -->wf 5354   ` cfv 5358  (class class class)co 5981   ↾t crest 13535   Topctop 16848  TopOnctopon 16849    Cn ccn 17171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-recs 6530  df-rdg 6565  df-oadd 6625  df-er 6802  df-map 6917  df-en 7007  df-fin 7010  df-fi 7312  df-rest 13537  df-topgen 13554  df-top 16853  df-bases 16855  df-topon 16856  df-cn 17174
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