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Theorem cnres2 29849
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 1019 . . 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 1017 . . 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 19545 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
51, 2, 4syl2anc 661 . 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 1016 . . . 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 19194 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
96, 8sylib 196 . . 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 5005 . . . 4  |-  ( F
" A )  =  ran  ( F  |`  A )
11 simp3r 1020 . . . . 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 19506 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> Y )
13 ffun 5724 . . . . . . 7  |-  ( F : X --> Y  ->  Fun  F )
141, 12, 133syl 20 . . . . . 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 5726 . . . . . . . 8  |-  ( F : X --> Y  ->  dom  F  =  X )
161, 12, 153syl 20 . . . . . . 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 3534 . . . . . 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 5909 . . . . . 6  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A. x  e.  A  ( F `  x )  e.  B ) )
1914, 17, 18syl2anc 661 . . . . 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 232 . . . 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 3542 . . 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 1018 . . 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 19546 . . 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 1223 . 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 210 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 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807    C_ wss 3469   U.cuni 4238   dom cdm 4992   ran crn 4993    |` cres 4994   "cima 4995   Fun wfun 5573   -->wf 5575   ` cfv 5579  (class class class)co 6275   ↾t crest 14665   Topctop 19154  TopOnctopon 19155    Cn ccn 19484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-fin 7510  df-fi 7860  df-rest 14667  df-topgen 14688  df-top 19159  df-bases 19161  df-topon 19162  df-cn 19487
This theorem is referenced by: (None)
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