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Theorem paste 23211
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 4212 . . . . . 6 (πœ‘ β†’ ((◑𝐹 β€œ 𝑦) ∩ (𝐴 βˆͺ 𝐡)) = ((◑𝐹 β€œ 𝑦) ∩ 𝑋))
4 indi 4274 . . . . . . 7 ((◑𝐹 β€œ 𝑦) ∩ (𝐴 βˆͺ 𝐡)) = (((◑𝐹 β€œ 𝑦) ∩ 𝐴) βˆͺ ((◑𝐹 β€œ 𝑦) ∩ 𝐡))
51ffund 6726 . . . . . . . 8 (πœ‘ β†’ Fun 𝐹)
6 respreima 7075 . . . . . . . . 9 (Fun 𝐹 β†’ (β—‘(𝐹 β†Ύ 𝐴) β€œ 𝑦) = ((◑𝐹 β€œ 𝑦) ∩ 𝐴))
7 respreima 7075 . . . . . . . . 9 (Fun 𝐹 β†’ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦) = ((◑𝐹 β€œ 𝑦) ∩ 𝐡))
86, 7uneq12d 4163 . . . . . . . 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 2787 . . . . . 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 6074 . . . . . . . . 9 (◑𝐹 β€œ 𝑦) βŠ† ran ◑𝐹
12 dfdm4 5898 . . . . . . . . . 10 dom 𝐹 = ran ◑𝐹
13 fdm 6731 . . . . . . . . . 10 (𝐹:π‘‹βŸΆπ‘Œ β†’ dom 𝐹 = 𝑋)
1412, 13eqtr3id 2782 . . . . . . . . 9 (𝐹:π‘‹βŸΆπ‘Œ β†’ ran ◑𝐹 = 𝑋)
1511, 14sseqtrid 4032 . . . . . . . 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 3964 . . . . . . 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 2777 . . . . 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 23185 . . . . . . 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 23091 . . . . . 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 684 . . . . 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 23185 . . . . . . 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 23091 . . . . . 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 684 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (Clsdβ€˜πΎ)) β†’ (β—‘(𝐹 β†Ύ 𝐡) β€œ 𝑦) ∈ (Clsdβ€˜π½))
33 uncld 22958 . . . . 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 2829 . . 3 ((πœ‘ ∧ 𝑦 ∈ (Clsdβ€˜πΎ)) β†’ (◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))
3635ralrimiva 3143 . 2 (πœ‘ β†’ βˆ€π‘¦ ∈ (Clsdβ€˜πΎ)(◑𝐹 β€œ 𝑦) ∈ (Clsdβ€˜π½))
37 cldrcl 22943 . . . 4 (𝐴 ∈ (Clsdβ€˜π½) β†’ 𝐽 ∈ Top)
3821, 37syl 17 . . 3 (πœ‘ β†’ 𝐽 ∈ Top)
39 cntop2 23158 . . . 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 22832 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
43 paste.2 . . . . 5 π‘Œ = βˆͺ 𝐾
4443toptopon 22832 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOnβ€˜π‘Œ))
45 iscncl 23186 . . . 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 712 1 (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058   βˆͺ cun 3945   ∩ cin 3946   βŠ† wss 3947  βˆͺ cuni 4908  β—‘ccnv 5677  dom cdm 5678  ran crn 5679   β†Ύ cres 5680   β€œ cima 5681  Fun wfun 6542  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420   β†Ύt crest 17402  Topctop 22808  TopOnctopon 22825  Clsdccld 22933   Cn ccn 23141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-map 8847  df-en 8965  df-fin 8968  df-fi 9435  df-rest 17404  df-topgen 17425  df-top 22809  df-topon 22826  df-bases 22862  df-cld 22936  df-cn 23144
This theorem is referenced by:  cnmpopc  24862  cvmliftlem10  34904
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