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Theorem paste 17038
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 3383 . . . . . 6  |-  ( ph  ->  ( ( `' F " y )  i^i  ( A  u.  B )
)  =  ( ( `' F " y )  i^i  X ) )
4 ffun 5407 . . . . . . . . 9  |-  ( F : X --> Y  ->  Fun  F )
51, 4syl 15 . . . . . . . 8  |-  ( ph  ->  Fun  F )
6 respreima 5670 . . . . . . . . 9  |-  ( Fun 
F  ->  ( `' ( F  |`  A )
" y )  =  ( ( `' F " y )  i^i  A
) )
7 respreima 5670 . . . . . . . . 9  |-  ( Fun 
F  ->  ( `' ( F  |`  B )
" y )  =  ( ( `' F " y )  i^i  B
) )
86, 7uneq12d 3343 . . . . . . . 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 3428 . . . . . . 7  |-  ( ( `' F " y )  i^i  ( A  u.  B ) )  =  ( ( ( `' F " y )  i^i  A )  u.  ( ( `' F " y )  i^i  B
) )
119, 10syl6reqr 2347 . . . . . 6  |-  ( ph  ->  ( ( `' F " y )  i^i  ( A  u.  B )
)  =  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) ) )
12 imassrn 5041 . . . . . . . . 9  |-  ( `' F " y ) 
C_  ran  `' F
13 dfdm4 4888 . . . . . . . . . 10  |-  dom  F  =  ran  `' F
14 fdm 5409 . . . . . . . . . 10  |-  ( F : X --> Y  ->  dom  F  =  X )
1513, 14syl5eqr 2342 . . . . . . . . 9  |-  ( F : X --> Y  ->  ran  `' F  =  X
)
1612, 15syl5sseq 3239 . . . . . . . 8  |-  ( F : X --> Y  -> 
( `' F "
y )  C_  X
)
171, 16syl 15 . . . . . . 7  |-  ( ph  ->  ( `' F "
y )  C_  X
)
18 df-ss 3179 . . . . . . 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 2337 . . . . 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 17013 . . . . . . 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 16921 . . . . . 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 17013 . . . . . . 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 16921 . . . . . 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 16794 . . . . 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 2370 . . 3  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' F " y )  e.  ( Clsd `  J
) )
3938ralrimiva 2639 . 2  |-  ( ph  ->  A. y  e.  (
Clsd `  K )
( `' F "
y )  e.  (
Clsd `  J )
)
40 cldrcl 16779 . . . 4  |-  ( A  e.  ( Clsd `  J
)  ->  J  e.  Top )
4122, 40syl 15 . . 3  |-  ( ph  ->  J  e.  Top )
42 cntop2 16987 . . . 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 16687 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
46 paste.2 . . . . 5  |-  Y  = 
U. K
4746toptopon 16687 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
48 iscncl 17014 . . . 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 1632    e. wcel 1696   A.wral 2556    u. cun 3163    i^i cin 3164    C_ wss 3165   U.cuni 3843   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874   ↾t crest 13341   Topctop 16647  TopOnctopon 16648   Clsdccld 16769    Cn ccn 16970
This theorem is referenced by:  cnmpt2pc  18442  cvmliftlem10  23840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-fin 6883  df-fi 7181  df-rest 13343  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cld 16772  df-cn 16973
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