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Theorem paste 19978
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 3638 . . . . . 6  |-  ( ph  ->  ( ( `' F " y )  i^i  ( A  u.  B )
)  =  ( ( `' F " y )  i^i  X ) )
4 ffun 5670 . . . . . . . . 9  |-  ( F : X --> Y  ->  Fun  F )
51, 4syl 17 . . . . . . . 8  |-  ( ph  ->  Fun  F )
6 respreima 5948 . . . . . . . . 9  |-  ( Fun 
F  ->  ( `' ( F  |`  A )
" y )  =  ( ( `' F " y )  i^i  A
) )
7 respreima 5948 . . . . . . . . 9  |-  ( Fun 
F  ->  ( `' ( F  |`  B )
" y )  =  ( ( `' F " y )  i^i  B
) )
86, 7uneq12d 3595 . . . . . . . 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 3693 . . . . . . 7  |-  ( ( `' F " y )  i^i  ( A  u.  B ) )  =  ( ( ( `' F " y )  i^i  A )  u.  ( ( `' F " y )  i^i  B
) )
119, 10syl6reqr 2460 . . . . . 6  |-  ( ph  ->  ( ( `' F " y )  i^i  ( A  u.  B )
)  =  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) ) )
12 imassrn 5287 . . . . . . . . 9  |-  ( `' F " y ) 
C_  ran  `' F
13 dfdm4 5135 . . . . . . . . . 10  |-  dom  F  =  ran  `' F
14 fdm 5672 . . . . . . . . . 10  |-  ( F : X --> Y  ->  dom  F  =  X )
1513, 14syl5eqr 2455 . . . . . . . . 9  |-  ( F : X --> Y  ->  ran  `' F  =  X
)
1612, 15syl5sseq 3487 . . . . . . . 8  |-  ( F : X --> Y  -> 
( `' F "
y )  C_  X
)
171, 16syl 17 . . . . . . 7  |-  ( ph  ->  ( `' F "
y )  C_  X
)
18 df-ss 3425 . . . . . . 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 2450 . . . . 5  |-  ( ph  ->  ( `' F "
y )  =  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) ) )
2120adantr 463 . . . 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 463 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  A  e.  ( Clsd `  J )
)
24 paste.8 . . . . . . 7  |-  ( ph  ->  ( F  |`  A )  e.  ( ( Jt  A )  Cn  K ) )
25 cnclima 19952 . . . . . . 7  |-  ( ( ( F  |`  A )  e.  ( ( Jt  A )  Cn  K )  /\  y  e.  (
Clsd `  K )
)  ->  ( `' ( F  |`  A )
" y )  e.  ( Clsd `  ( Jt  A ) ) )
2624, 25sylan 469 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  A )
" y )  e.  ( Clsd `  ( Jt  A ) ) )
27 restcldr 19858 . . . . . 6  |-  ( ( A  e.  ( Clsd `  J )  /\  ( `' ( F  |`  A ) " y
)  e.  ( Clsd `  ( Jt  A ) ) )  ->  ( `' ( F  |`  A ) " y )  e.  ( Clsd `  J
) )
2823, 26, 27syl2anc 659 . . . . 5  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  A )
" y )  e.  ( Clsd `  J
) )
29 paste.5 . . . . . . 7  |-  ( ph  ->  B  e.  ( Clsd `  J ) )
3029adantr 463 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  B  e.  ( Clsd `  J )
)
31 paste.9 . . . . . . 7  |-  ( ph  ->  ( F  |`  B )  e.  ( ( Jt  B )  Cn  K ) )
32 cnclima 19952 . . . . . . 7  |-  ( ( ( F  |`  B )  e.  ( ( Jt  B )  Cn  K )  /\  y  e.  (
Clsd `  K )
)  ->  ( `' ( F  |`  B )
" y )  e.  ( Clsd `  ( Jt  B ) ) )
3331, 32sylan 469 . . . . . 6  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  B )
" y )  e.  ( Clsd `  ( Jt  B ) ) )
34 restcldr 19858 . . . . . 6  |-  ( ( B  e.  ( Clsd `  J )  /\  ( `' ( F  |`  B ) " y
)  e.  ( Clsd `  ( Jt  B ) ) )  ->  ( `' ( F  |`  B ) " y )  e.  ( Clsd `  J
) )
3530, 33, 34syl2anc 659 . . . . 5  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' ( F  |`  B )
" y )  e.  ( Clsd `  J
) )
36 uncld 19724 . . . . 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 659 . . . 4  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( ( `' ( F  |`  A ) " y
)  u.  ( `' ( F  |`  B )
" y ) )  e.  ( Clsd `  J
) )
3821, 37eqeltrd 2488 . . 3  |-  ( (
ph  /\  y  e.  ( Clsd `  K )
)  ->  ( `' F " y )  e.  ( Clsd `  J
) )
3938ralrimiva 2815 . 2  |-  ( ph  ->  A. y  e.  (
Clsd `  K )
( `' F "
y )  e.  (
Clsd `  J )
)
40 cldrcl 19709 . . . 4  |-  ( A  e.  ( Clsd `  J
)  ->  J  e.  Top )
4122, 40syl 17 . . 3  |-  ( ph  ->  J  e.  Top )
42 cntop2 19925 . . . 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 19616 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
46 paste.2 . . . . 5  |-  Y  = 
U. K
4746toptopon 19616 . . . 4  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
48 iscncl 19953 . . . 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 477 . . 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 659 . 2  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> Y  /\  A. y  e.  ( Clsd `  K
) ( `' F " y )  e.  (
Clsd `  J )
) ) )
511, 39, 50mpbir2and 921 1  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1403    e. wcel 1840   A.wral 2751    u. cun 3409    i^i cin 3410    C_ wss 3411   U.cuni 4188   `'ccnv 4939   dom cdm 4940   ran crn 4941    |` cres 4942   "cima 4943   Fun wfun 5517   -->wf 5519   ` cfv 5523  (class class class)co 6232   ↾t crest 14925   Topctop 19576  TopOnctopon 19577   Clsdccld 19699    Cn ccn 19908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-oadd 7089  df-er 7266  df-map 7377  df-en 7473  df-fin 7476  df-fi 7823  df-rest 14927  df-topgen 14948  df-top 19581  df-bases 19583  df-topon 19584  df-cld 19702  df-cn 19911
This theorem is referenced by:  cnmpt2pc  21610  cvmliftlem10  29467
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