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Theorem paste 17022
Description: Pasting lemma. If  A and  B are closed sets in  X with  A  u.  B  =  X, then any function whose restrictions to  A and  B are continuous is continuous on all of  X. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
Hypotheses
Ref Expression
paste.1  |-  X  = 
U. J
paste.2  |-  Y  = 
U. K
paste.4  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
paste.5  |-  ( ph  ->  B  e.  ( Clsd `  J ) )
paste.6  |-  ( ph  ->  ( A  u.  B
)  =  X )
paste.7  |-  ( ph  ->  F : X --> Y )
paste.8  |-  ( ph  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
paste.9  |-  ( ph  ->  ( F  |`  B )  e.  ( ( Jt  B )  Cn  K ) )
Assertion
Ref Expression
paste  |-  ( ph  ->  F  e.  ( J  Cn  K ) )

Proof of Theorem paste
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 paste.7 . 2  |-  ( ph  ->  F : X --> Y )
2 paste.6 . . . . . . 7  |-  ( ph  ->  ( A  u.  B
)  =  X )
32ineq2d 3370 . . . . . 6  |-  ( ph  ->  ( ( `' F " y )  i^i  ( A  u.  B )
)  =  ( ( `' F " y )  i^i  X ) )
4 ffun 5391 . . . . . . . . 9  |-  ( F : X --> Y  ->  Fun  F )
51, 4syl 15 . . . . . . . 8  |-  ( ph  ->  Fun  F )
6 respreima 5654 . . . . . . . . 9  |-  ( Fun 
F  ->  ( `' ( F  |`  A )
" y )  =  ( ( `' F " y )  i^i  A
) )
7 respreima 5654 . . . . . . . . 9  |-  ( Fun 
F  ->  ( `' ( F  |`  B )
" y )  =  ( ( `' F " y )  i^i  B
) )
86, 7uneq12d 3330 . . . . . . . 8  |-  ( Fun 
F  ->  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) )  =  ( ( ( `' F " y )  i^i  A )  u.  ( ( `' F " y )  i^i  B
) ) )
95, 8syl 15 . . . . . . 7  |-  ( ph  ->  ( ( `' ( F  |`  A ) " y )  u.  ( `' ( F  |`  B ) " y
) )  =  ( ( ( `' F " y )  i^i  A
)  u.  ( ( `' F " y )  i^i  B ) ) )
10 indi 3415 . . . . . . 7  |-  ( ( `' F " y )  i^i  ( A  u.  B ) )  =  ( ( ( `' F " y )  i^i  A )  u.  ( ( `' F " y )  i^i  B
) )
119, 10syl6reqr 2334 . . . . . 6  |-  ( ph  ->  ( ( `' F " y )  i^i  ( A  u.  B )
)  =  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) ) )
12 imassrn 5025 . . . . . . . . 9  |-  ( `' F " y ) 
C_  ran  `' F
13 dfdm4 4872 . . . . . . . . . 10  |-  dom  F  =  ran  `' F
14 fdm 5393 . . . . . . . . . 10  |-  ( F : X --> Y  ->  dom  F  =  X )
1513, 14syl5eqr 2329 . . . . . . . . 9  |-  ( F : X --> Y  ->  ran  `' F  =  X
)
1612, 15syl5sseq 3226 . . . . . . . 8  |-  ( F : X --> Y  -> 
( `' F "
y )  C_  X
)
171, 16syl 15 . . . . . . 7  |-  ( ph  ->  ( `' F "
y )  C_  X
)
18 df-ss 3166 . . . . . . 7  |-  ( ( `' F " y ) 
C_  X  <->  ( ( `' F " y )  i^i  X )  =  ( `' F "
y ) )
1917, 18sylib 188 . . . . . 6  |-  ( ph  ->  ( ( `' F " y )  i^i  X
)  =  ( `' F " y ) )
203, 11, 193eqtr3rd 2324 . . . . 5  |-  ( ph  ->  ( `' F "
y )  =  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) ) )
2120adantr 451 . . . 4  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' F " y )  =  ( ( `' ( F  |`  A ) " y )  u.  ( `' ( F  |`  B ) " y
) ) )
22 paste.4 . . . . . . 7  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
2322adantr 451 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  A  e.  ( Clsd `  J )
)
24 paste.8 . . . . . . 7  |-  ( ph  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
25 cnclima 16997 . . . . . . 7  |-  ( ( ( F  |`  A )  e.  ( ( Jt  A )  Cn  K )  /\  y  e.  (
Clsd `  K )
)  ->  ( `' ( F  |`  A )
" y )  e.  ( Clsd `  ( Jt  A ) ) )
2624, 25sylan 457 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  A )
" y )  e.  ( Clsd `  ( Jt  A ) ) )
27 restcldr 16905 . . . . . 6  |-  ( ( A  e.  ( Clsd `  J )  /\  ( `' ( F  |`  A ) " y
)  e.  ( Clsd `  ( Jt  A ) ) )  ->  ( `' ( F  |`  A ) " y )  e.  ( Clsd `  J
) )
2823, 26, 27syl2anc 642 . . . . 5  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  A )
" y )  e.  ( Clsd `  J
) )
29 paste.5 . . . . . . 7  |-  ( ph  ->  B  e.  ( Clsd `  J ) )
3029adantr 451 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  B  e.  ( Clsd `  J )
)
31 paste.9 . . . . . . 7  |-  ( ph  ->  ( F  |`  B )  e.  ( ( Jt  B )  Cn  K ) )
32 cnclima 16997 . . . . . . 7  |-  ( ( ( F  |`  B )  e.  ( ( Jt  B )  Cn  K )  /\  y  e.  (
Clsd `  K )
)  ->  ( `' ( F  |`  B )
" y )  e.  ( Clsd `  ( Jt  B ) ) )
3331, 32sylan 457 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  B )
" y )  e.  ( Clsd `  ( Jt  B ) ) )
34 restcldr 16905 . . . . . 6  |-  ( ( B  e.  ( Clsd `  J )  /\  ( `' ( F  |`  B ) " y
)  e.  ( Clsd `  ( Jt  B ) ) )  ->  ( `' ( F  |`  B ) " y )  e.  ( Clsd `  J
) )
3530, 33, 34syl2anc 642 . . . . 5  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  B )
" y )  e.  ( Clsd `  J
) )
36 uncld 16778 . . . . 5  |-  ( ( ( `' ( F  |`  A ) " y
)  e.  ( Clsd `  J )  /\  ( `' ( F  |`  B ) " y
)  e.  ( Clsd `  J ) )  -> 
( ( `' ( F  |`  A ) " y )  u.  ( `' ( F  |`  B ) " y
) )  e.  (
Clsd `  J )
)
3728, 35, 36syl2anc 642 . . . 4  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) )  e.  ( Clsd `  J
) )
3821, 37eqeltrd 2357 . . 3  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' F " y )  e.  ( Clsd `  J
) )
3938ralrimiva 2626 . 2  |-  ( ph  ->  A. y  e.  (
Clsd `  K )
( `' F "
y )  e.  (
Clsd `  J )
)
40 cldrcl 16763 . . . 4  |-  ( A  e.  ( Clsd `  J
)  ->  J  e.  Top )
4122, 40syl 15 . . 3  |-  ( ph  ->  J  e.  Top )
42 cntop2 16971 . . . 4  |-  ( ( F  |`  A )  e.  ( ( Jt  A )  Cn  K )  ->  K  e.  Top )
4324, 42syl 15 . . 3  |-  ( ph  ->  K  e.  Top )
44 paste.1 . . . . 5  |-  X  = 
U. J
4544toptopon 16671 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
46 paste.2 . . . . 5  |-  Y  = 
U. K
4746toptopon 16671 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
48 iscncl 16998 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> Y  /\  A. y  e.  ( Clsd `  K
) ( `' F " y )  e.  (
Clsd `  J )
) ) )
4945, 47, 48syl2anb 465 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> Y  /\  A. y  e.  ( Clsd `  K
) ( `' F " y )  e.  (
Clsd `  J )
) ) )
5041, 43, 49syl2anc 642 . 2  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> Y  /\  A. y  e.  ( Clsd `  K
) ( `' F " y )  e.  (
Clsd `  J )
) ) )
511, 39, 50mpbir2and 888 1  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    u. cun 3150    i^i cin 3151    C_ wss 3152   U.cuni 3827   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631  TopOnctopon 16632   Clsdccld 16753    Cn ccn 16954
This theorem is referenced by:  cnmpt2pc  18426  cvmliftlem10  23825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-fin 6867  df-fi 7165  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-cn 16957
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