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Theorem paste 20387
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 3625 . . . . . 6  |-  ( ph  ->  ( ( `' F " y )  i^i  ( A  u.  B )
)  =  ( ( `' F " y )  i^i  X ) )
4 ffun 5742 . . . . . . . . 9  |-  ( F : X --> Y  ->  Fun  F )
51, 4syl 17 . . . . . . . 8  |-  ( ph  ->  Fun  F )
6 respreima 6024 . . . . . . . . 9  |-  ( Fun 
F  ->  ( `' ( F  |`  A )
" y )  =  ( ( `' F " y )  i^i  A
) )
7 respreima 6024 . . . . . . . . 9  |-  ( Fun 
F  ->  ( `' ( F  |`  B )
" y )  =  ( ( `' F " y )  i^i  B
) )
86, 7uneq12d 3580 . . . . . . . 8  |-  ( Fun 
F  ->  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) )  =  ( ( ( `' F " y )  i^i  A )  u.  ( ( `' F " y )  i^i  B
) ) )
95, 8syl 17 . . . . . . 7  |-  ( ph  ->  ( ( `' ( F  |`  A ) " y )  u.  ( `' ( F  |`  B ) " y
) )  =  ( ( ( `' F " y )  i^i  A
)  u.  ( ( `' F " y )  i^i  B ) ) )
10 indi 3680 . . . . . . 7  |-  ( ( `' F " y )  i^i  ( A  u.  B ) )  =  ( ( ( `' F " y )  i^i  A )  u.  ( ( `' F " y )  i^i  B
) )
119, 10syl6reqr 2524 . . . . . 6  |-  ( ph  ->  ( ( `' F " y )  i^i  ( A  u.  B )
)  =  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) ) )
12 imassrn 5185 . . . . . . . . 9  |-  ( `' F " y ) 
C_  ran  `' F
13 dfdm4 5032 . . . . . . . . . 10  |-  dom  F  =  ran  `' F
14 fdm 5745 . . . . . . . . . 10  |-  ( F : X --> Y  ->  dom  F  =  X )
1513, 14syl5eqr 2519 . . . . . . . . 9  |-  ( F : X --> Y  ->  ran  `' F  =  X
)
1612, 15syl5sseq 3466 . . . . . . . 8  |-  ( F : X --> Y  -> 
( `' F "
y )  C_  X
)
171, 16syl 17 . . . . . . 7  |-  ( ph  ->  ( `' F "
y )  C_  X
)
18 df-ss 3404 . . . . . . 7  |-  ( ( `' F " y ) 
C_  X  <->  ( ( `' F " y )  i^i  X )  =  ( `' F "
y ) )
1917, 18sylib 201 . . . . . 6  |-  ( ph  ->  ( ( `' F " y )  i^i  X
)  =  ( `' F " y ) )
203, 11, 193eqtr3rd 2514 . . . . 5  |-  ( ph  ->  ( `' F "
y )  =  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) ) )
2120adantr 472 . . . 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 472 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  A  e.  ( Clsd `  J )
)
24 paste.8 . . . . . . 7  |-  ( ph  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
25 cnclima 20361 . . . . . . 7  |-  ( ( ( F  |`  A )  e.  ( ( Jt  A )  Cn  K )  /\  y  e.  (
Clsd `  K )
)  ->  ( `' ( F  |`  A )
" y )  e.  ( Clsd `  ( Jt  A ) ) )
2624, 25sylan 479 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  A )
" y )  e.  ( Clsd `  ( Jt  A ) ) )
27 restcldr 20267 . . . . . 6  |-  ( ( A  e.  ( Clsd `  J )  /\  ( `' ( F  |`  A ) " y
)  e.  ( Clsd `  ( Jt  A ) ) )  ->  ( `' ( F  |`  A ) " y )  e.  ( Clsd `  J
) )
2823, 26, 27syl2anc 673 . . . . 5  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  A )
" y )  e.  ( Clsd `  J
) )
29 paste.5 . . . . . . 7  |-  ( ph  ->  B  e.  ( Clsd `  J ) )
3029adantr 472 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  B  e.  ( Clsd `  J )
)
31 paste.9 . . . . . . 7  |-  ( ph  ->  ( F  |`  B )  e.  ( ( Jt  B )  Cn  K ) )
32 cnclima 20361 . . . . . . 7  |-  ( ( ( F  |`  B )  e.  ( ( Jt  B )  Cn  K )  /\  y  e.  (
Clsd `  K )
)  ->  ( `' ( F  |`  B )
" y )  e.  ( Clsd `  ( Jt  B ) ) )
3331, 32sylan 479 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  B )
" y )  e.  ( Clsd `  ( Jt  B ) ) )
34 restcldr 20267 . . . . . 6  |-  ( ( B  e.  ( Clsd `  J )  /\  ( `' ( F  |`  B ) " y
)  e.  ( Clsd `  ( Jt  B ) ) )  ->  ( `' ( F  |`  B ) " y )  e.  ( Clsd `  J
) )
3530, 33, 34syl2anc 673 . . . . 5  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  B )
" y )  e.  ( Clsd `  J
) )
36 uncld 20133 . . . . 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 673 . . . 4  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) )  e.  ( Clsd `  J
) )
3821, 37eqeltrd 2549 . . 3  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' F " y )  e.  ( Clsd `  J
) )
3938ralrimiva 2809 . 2  |-  ( ph  ->  A. y  e.  (
Clsd `  K )
( `' F "
y )  e.  (
Clsd `  J )
)
40 cldrcl 20118 . . . 4  |-  ( A  e.  ( Clsd `  J
)  ->  J  e.  Top )
4122, 40syl 17 . . 3  |-  ( ph  ->  J  e.  Top )
42 cntop2 20334 . . . 4  |-  ( ( F  |`  A )  e.  ( ( Jt  A )  Cn  K )  ->  K  e.  Top )
4324, 42syl 17 . . 3  |-  ( ph  ->  K  e.  Top )
44 paste.1 . . . . 5  |-  X  = 
U. J
4544toptopon 20025 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
46 paste.2 . . . . 5  |-  Y  = 
U. K
4746toptopon 20025 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
48 iscncl 20362 . . . 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 487 . . 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 673 . 2  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> Y  /\  A. y  e.  ( Clsd `  K
) ( `' F " y )  e.  (
Clsd `  J )
) ) )
511, 39, 50mpbir2and 936 1  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756    u. cun 3388    i^i cin 3389    C_ wss 3390   U.cuni 4190   `'ccnv 4838   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   ↾t crest 15397   Topctop 19994  TopOnctopon 19995   Clsdccld 20108    Cn ccn 20317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-fin 7591  df-fi 7943  df-rest 15399  df-topgen 15420  df-top 19998  df-bases 19999  df-topon 20000  df-cld 20111  df-cn 20320
This theorem is referenced by:  cnmpt2pc  22034  cvmliftlem10  30089
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