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Theorem paste 19033
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 3663 . . . . . 6  |-  ( ph  ->  ( ( `' F " y )  i^i  ( A  u.  B )
)  =  ( ( `' F " y )  i^i  X ) )
4 ffun 5672 . . . . . . . . 9  |-  ( F : X --> Y  ->  Fun  F )
51, 4syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  F )
6 respreima 5944 . . . . . . . . 9  |-  ( Fun 
F  ->  ( `' ( F  |`  A )
" y )  =  ( ( `' F " y )  i^i  A
) )
7 respreima 5944 . . . . . . . . 9  |-  ( Fun 
F  ->  ( `' ( F  |`  B )
" y )  =  ( ( `' F " y )  i^i  B
) )
86, 7uneq12d 3622 . . . . . . . 8  |-  ( Fun 
F  ->  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) )  =  ( ( ( `' F " y )  i^i  A )  u.  ( ( `' F " y )  i^i  B
) ) )
95, 8syl 16 . . . . . . 7  |-  ( ph  ->  ( ( `' ( F  |`  A ) " y )  u.  ( `' ( F  |`  B ) " y
) )  =  ( ( ( `' F " y )  i^i  A
)  u.  ( ( `' F " y )  i^i  B ) ) )
10 indi 3707 . . . . . . 7  |-  ( ( `' F " y )  i^i  ( A  u.  B ) )  =  ( ( ( `' F " y )  i^i  A )  u.  ( ( `' F " y )  i^i  B
) )
119, 10syl6reqr 2514 . . . . . 6  |-  ( ph  ->  ( ( `' F " y )  i^i  ( A  u.  B )
)  =  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) ) )
12 imassrn 5291 . . . . . . . . 9  |-  ( `' F " y ) 
C_  ran  `' F
13 dfdm4 5143 . . . . . . . . . 10  |-  dom  F  =  ran  `' F
14 fdm 5674 . . . . . . . . . 10  |-  ( F : X --> Y  ->  dom  F  =  X )
1513, 14syl5eqr 2509 . . . . . . . . 9  |-  ( F : X --> Y  ->  ran  `' F  =  X
)
1612, 15syl5sseq 3515 . . . . . . . 8  |-  ( F : X --> Y  -> 
( `' F "
y )  C_  X
)
171, 16syl 16 . . . . . . 7  |-  ( ph  ->  ( `' F "
y )  C_  X
)
18 df-ss 3453 . . . . . . 7  |-  ( ( `' F " y ) 
C_  X  <->  ( ( `' F " y )  i^i  X )  =  ( `' F "
y ) )
1917, 18sylib 196 . . . . . 6  |-  ( ph  ->  ( ( `' F " y )  i^i  X
)  =  ( `' F " y ) )
203, 11, 193eqtr3rd 2504 . . . . 5  |-  ( ph  ->  ( `' F "
y )  =  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) ) )
2120adantr 465 . . . 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 465 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  A  e.  ( Clsd `  J )
)
24 paste.8 . . . . . . 7  |-  ( ph  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
25 cnclima 19007 . . . . . . 7  |-  ( ( ( F  |`  A )  e.  ( ( Jt  A )  Cn  K )  /\  y  e.  (
Clsd `  K )
)  ->  ( `' ( F  |`  A )
" y )  e.  ( Clsd `  ( Jt  A ) ) )
2624, 25sylan 471 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  A )
" y )  e.  ( Clsd `  ( Jt  A ) ) )
27 restcldr 18913 . . . . . 6  |-  ( ( A  e.  ( Clsd `  J )  /\  ( `' ( F  |`  A ) " y
)  e.  ( Clsd `  ( Jt  A ) ) )  ->  ( `' ( F  |`  A ) " y )  e.  ( Clsd `  J
) )
2823, 26, 27syl2anc 661 . . . . 5  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  A )
" y )  e.  ( Clsd `  J
) )
29 paste.5 . . . . . . 7  |-  ( ph  ->  B  e.  ( Clsd `  J ) )
3029adantr 465 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  B  e.  ( Clsd `  J )
)
31 paste.9 . . . . . . 7  |-  ( ph  ->  ( F  |`  B )  e.  ( ( Jt  B )  Cn  K ) )
32 cnclima 19007 . . . . . . 7  |-  ( ( ( F  |`  B )  e.  ( ( Jt  B )  Cn  K )  /\  y  e.  (
Clsd `  K )
)  ->  ( `' ( F  |`  B )
" y )  e.  ( Clsd `  ( Jt  B ) ) )
3331, 32sylan 471 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  B )
" y )  e.  ( Clsd `  ( Jt  B ) ) )
34 restcldr 18913 . . . . . 6  |-  ( ( B  e.  ( Clsd `  J )  /\  ( `' ( F  |`  B ) " y
)  e.  ( Clsd `  ( Jt  B ) ) )  ->  ( `' ( F  |`  B ) " y )  e.  ( Clsd `  J
) )
3530, 33, 34syl2anc 661 . . . . 5  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  B )
" y )  e.  ( Clsd `  J
) )
36 uncld 18780 . . . . 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 661 . . . 4  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) )  e.  ( Clsd `  J
) )
3821, 37eqeltrd 2542 . . 3  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' F " y )  e.  ( Clsd `  J
) )
3938ralrimiva 2830 . 2  |-  ( ph  ->  A. y  e.  (
Clsd `  K )
( `' F "
y )  e.  (
Clsd `  J )
)
40 cldrcl 18765 . . . 4  |-  ( A  e.  ( Clsd `  J
)  ->  J  e.  Top )
4122, 40syl 16 . . 3  |-  ( ph  ->  J  e.  Top )
42 cntop2 18980 . . . 4  |-  ( ( F  |`  A )  e.  ( ( Jt  A )  Cn  K )  ->  K  e.  Top )
4324, 42syl 16 . . 3  |-  ( ph  ->  K  e.  Top )
44 paste.1 . . . . 5  |-  X  = 
U. J
4544toptopon 18673 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
46 paste.2 . . . . 5  |-  Y  = 
U. K
4746toptopon 18673 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
48 iscncl 19008 . . . 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 479 . . 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 661 . 2  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> Y  /\  A. y  e.  ( Clsd `  K
) ( `' F " y )  e.  (
Clsd `  J )
) ) )
511, 39, 50mpbir2and 913 1  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799    u. cun 3437    i^i cin 3438    C_ wss 3439   U.cuni 4202   `'ccnv 4950   dom cdm 4951   ran crn 4952    |` cres 4953   "cima 4954   Fun wfun 5523   -->wf 5525   ` cfv 5529  (class class class)co 6203   ↾t crest 14481   Topctop 18633  TopOnctopon 18634   Clsdccld 18755    Cn ccn 18963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-fin 7427  df-fi 7775  df-rest 14483  df-topgen 14504  df-top 18638  df-bases 18640  df-topon 18641  df-cld 18758  df-cn 18966
This theorem is referenced by:  cnmpt2pc  20635  cvmliftlem10  27347
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