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Theorem paste 23142
Description: Pasting lemma. If 𝐴 and 𝐡 are closed sets in 𝑋 with 𝐴 βˆͺ 𝐡 = 𝑋, then any function whose restrictions to 𝐴 and 𝐡 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 = βˆͺ 𝐽
paste.2 π‘Œ = βˆͺ 𝐾
paste.4 (πœ‘ β†’ 𝐴 ∈ (Clsdβ€˜π½))
paste.5 (πœ‘ β†’ 𝐡 ∈ (Clsdβ€˜π½))
paste.6 (πœ‘ β†’ (𝐴 βˆͺ 𝐡) = 𝑋)
paste.7 (πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘Œ)
paste.8 (πœ‘ β†’ (𝐹 β†Ύ 𝐴) ∈ ((𝐽 β†Ύt 𝐴) Cn 𝐾))
paste.9 (πœ‘ β†’ (𝐹 β†Ύ 𝐡) ∈ ((𝐽 β†Ύt 𝐡) Cn 𝐾))
Assertion
Ref Expression
paste (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
Proof of Theorem paste
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 paste.7 . 2 (πœ‘ β†’ 𝐹:π‘‹βŸΆπ‘Œ)
2 paste.6 . . . . . . 7 (πœ‘ β†’ (𝐴 βˆͺ 𝐡) = 𝑋)
32ineq2d 4205 . . . . . 6 (πœ‘ β†’ ((◑𝐹 β€œ 𝑦) ∩ (𝐴 βˆͺ 𝐡)) = ((◑𝐹 β€œ 𝑦) ∩ 𝑋))
4 indi 4266 . . . . . . 7 ((◑𝐹 β€œ 𝑦) ∩ (𝐴 βˆͺ 𝐡)) = (((◑𝐹 β€œ 𝑦) ∩ 𝐴) βˆͺ ((◑𝐹 β€œ 𝑦) ∩ 𝐡))
51ffund 6712 . . . . . . . 8 (πœ‘ β†’ Fun 𝐹)
6 respreima 7058 . . . . . . . . 9 (Fun 𝐹 β†’ (β—‘(𝐹 β†Ύ 𝐴) β€œ 𝑦) = ((◑𝐹 β€œ 𝑦) ∩ 𝐴))
7 respreima 7058 . . . . . . . . 9 (Fun 𝐹 β†’ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦) = ((◑𝐹 β€œ 𝑦) ∩ 𝐡))
86, 7uneq12d 4157 . . . . . . . 8 (Fun 𝐹 β†’ ((β—‘(𝐹 β†Ύ 𝐴) β€œ 𝑦) βˆͺ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦)) = (((◑𝐹 β€œ 𝑦) ∩ 𝐴) βˆͺ ((◑𝐹 β€œ 𝑦) ∩ 𝐡)))
95, 8syl 17 . . . . . . 7 (πœ‘ β†’ ((β—‘(𝐹 β†Ύ 𝐴) β€œ 𝑦) βˆͺ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦)) = (((◑𝐹 β€œ 𝑦) ∩ 𝐴) βˆͺ ((◑𝐹 β€œ 𝑦) ∩ 𝐡)))
104, 9eqtr4id 2783 . . . . . 6 (πœ‘ β†’ ((◑𝐹 β€œ 𝑦) ∩ (𝐴 βˆͺ 𝐡)) = ((β—‘(𝐹 β†Ύ 𝐴) β€œ 𝑦) βˆͺ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦)))
11 imassrn 6061 . . . . . . . . 9 (◑𝐹 β€œ 𝑦) βŠ† ran ◑𝐹
12 dfdm4 5886 . . . . . . . . . 10 dom 𝐹 = ran ◑𝐹
13 fdm 6717 . . . . . . . . . 10 (𝐹:π‘‹βŸΆπ‘Œ β†’ dom 𝐹 = 𝑋)
1412, 13eqtr3id 2778 . . . . . . . . 9 (𝐹:π‘‹βŸΆπ‘Œ β†’ ran ◑𝐹 = 𝑋)
1511, 14sseqtrid 4027 . . . . . . . 8 (𝐹:π‘‹βŸΆπ‘Œ β†’ (◑𝐹 β€œ 𝑦) βŠ† 𝑋)
161, 15syl 17 . . . . . . 7 (πœ‘ β†’ (◑𝐹 β€œ 𝑦) βŠ† 𝑋)
17 df-ss 3958 . . . . . . 7 ((◑𝐹 β€œ 𝑦) βŠ† 𝑋 ↔ ((◑𝐹 β€œ 𝑦) ∩ 𝑋) = (◑𝐹 β€œ 𝑦))
1816, 17sylib 217 . . . . . 6 (πœ‘ β†’ ((◑𝐹 β€œ 𝑦) ∩ 𝑋) = (◑𝐹 β€œ 𝑦))
193, 10, 183eqtr3rd 2773 . . . . 5 (πœ‘ β†’ (◑𝐹 β€œ 𝑦) = ((β—‘(𝐹 β†Ύ 𝐴) β€œ 𝑦) βˆͺ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦)))
2019adantr 480 . . . 4 ((πœ‘ ∧ 𝑦 ∈ (Clsdβ€˜πΎ)) β†’ (◑𝐹 β€œ 𝑦) = ((β—‘(𝐹 β†Ύ 𝐴) β€œ 𝑦) βˆͺ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦)))
21 paste.4 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ (Clsdβ€˜π½))
22 paste.8 . . . . . . 7 (πœ‘ β†’ (𝐹 β†Ύ 𝐴) ∈ ((𝐽 β†Ύt 𝐴) Cn 𝐾))
23 cnclima 23116 . . . . . . 7 (((𝐹 β†Ύ 𝐴) ∈ ((𝐽 β†Ύt 𝐴) Cn 𝐾) ∧ 𝑦 ∈ (Clsdβ€˜πΎ)) β†’ (β—‘(𝐹 β†Ύ 𝐴) β€œ 𝑦) ∈ (Clsdβ€˜(𝐽 β†Ύt 𝐴)))
2422, 23sylan 579 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ (Clsdβ€˜πΎ)) β†’ (β—‘(𝐹 β†Ύ 𝐴) β€œ 𝑦) ∈ (Clsdβ€˜(𝐽 β†Ύt 𝐴)))
25 restcldr 23022 . . . . . 6 ((𝐴 ∈ (Clsdβ€˜π½) ∧ (β—‘(𝐹 β†Ύ 𝐴) β€œ 𝑦) ∈ (Clsdβ€˜(𝐽 β†Ύt 𝐴))) β†’ (β—‘(𝐹 β†Ύ 𝐴) β€œ 𝑦) ∈ (Clsdβ€˜π½))
2621, 24, 25syl2an2r 682 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (Clsdβ€˜πΎ)) β†’ (β—‘(𝐹 β†Ύ 𝐴) β€œ 𝑦) ∈ (Clsdβ€˜π½))
27 paste.5 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ (Clsdβ€˜π½))
28 paste.9 . . . . . . 7 (πœ‘ β†’ (𝐹 β†Ύ 𝐡) ∈ ((𝐽 β†Ύt 𝐡) Cn 𝐾))
29 cnclima 23116 . . . . . . 7 (((𝐹 β†Ύ 𝐡) ∈ ((𝐽 β†Ύt 𝐡) Cn 𝐾) ∧ 𝑦 ∈ (Clsdβ€˜πΎ)) β†’ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦) ∈ (Clsdβ€˜(𝐽 β†Ύt 𝐡)))
3028, 29sylan 579 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ (Clsdβ€˜πΎ)) β†’ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦) ∈ (Clsdβ€˜(𝐽 β†Ύt 𝐡)))
31 restcldr 23022 . . . . . 6 ((𝐡 ∈ (Clsdβ€˜π½) ∧ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦) ∈ (Clsdβ€˜(𝐽 β†Ύt 𝐡))) β†’ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦) ∈ (Clsdβ€˜π½))
3227, 30, 31syl2an2r 682 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (Clsdβ€˜πΎ)) β†’ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦) ∈ (Clsdβ€˜π½))
33 uncld 22889 . . . . 5 (((β—‘(𝐹 β†Ύ 𝐴) β€œ 𝑦) ∈ (Clsdβ€˜π½) ∧ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦) ∈ (Clsdβ€˜π½)) β†’ ((β—‘(𝐹 β†Ύ 𝐴) β€œ 𝑦) βˆͺ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦)) ∈ (Clsdβ€˜π½))
3426, 32, 33syl2anc 583 . . . 4 ((πœ‘ ∧ 𝑦 ∈ (Clsdβ€˜πΎ)) β†’ ((β—‘(𝐹 β†Ύ 𝐴) β€œ 𝑦) βˆͺ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦)) ∈ (Clsdβ€˜π½))
3520, 34eqeltrd 2825 . . 3 ((πœ‘ ∧ 𝑦 ∈ (Clsdβ€˜πΎ)) β†’ (◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))
3635ralrimiva 3138 . 2 (πœ‘ β†’ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))
37 cldrcl 22874 . . . 4 (𝐴 ∈ (Clsdβ€˜π½) β†’ 𝐽 ∈ Top)
3821, 37syl 17 . . 3 (πœ‘ β†’ 𝐽 ∈ Top)
39 cntop2 23089 . . . 4 ((𝐹 β†Ύ 𝐴) ∈ ((𝐽 β†Ύt 𝐴) Cn 𝐾) β†’ 𝐾 ∈ Top)
4022, 39syl 17 . . 3 (πœ‘ β†’ 𝐾 ∈ Top)
41 paste.1 . . . . 5 𝑋 = βˆͺ 𝐽
4241toptopon 22763 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
43 paste.2 . . . . 5 π‘Œ = βˆͺ 𝐾
4443toptopon 22763 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜π‘Œ))
45 iscncl 23117 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))))
4642, 44, 45syl2anb 597 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))))
4738, 40, 46syl2anc 583 . 2 (πœ‘ β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))))
481, 36, 47mpbir2and 710 1 (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053   βˆͺ cun 3939   ∩ cin 3940   βŠ† wss 3941  βˆͺ cuni 4900  β—‘ccnv 5666  dom cdm 5667  ran crn 5668   β†Ύ cres 5669   β€œ cima 5670  Fun wfun 6528  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402   β†Ύt crest 17371  Topctop 22739  TopOnctopon 22756  Clsdccld 22864   Cn ccn 23072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-map 8819  df-en 8937  df-fin 8940  df-fi 9403  df-rest 17373  df-topgen 17394  df-top 22740  df-topon 22757  df-bases 22793  df-cld 22867  df-cn 23075
This theorem is referenced by:  cnmpopc  24793  cvmliftlem10  34803
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