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Theorem paste 21901
 Description: Pasting lemma. If 𝐴 and 𝐵 are closed sets in 𝑋 with 𝐴 ∪ 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 = 𝐽
paste.2 𝑌 = 𝐾
paste.4 (𝜑𝐴 ∈ (Clsd‘𝐽))
paste.5 (𝜑𝐵 ∈ (Clsd‘𝐽))
paste.6 (𝜑 → (𝐴𝐵) = 𝑋)
paste.7 (𝜑𝐹:𝑋𝑌)
paste.8 (𝜑 → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾))
paste.9 (𝜑 → (𝐹𝐵) ∈ ((𝐽t 𝐵) Cn 𝐾))
Assertion
Ref Expression
paste (𝜑𝐹 ∈ (𝐽 Cn 𝐾))

Proof of Theorem paste
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 paste.7 . 2 (𝜑𝐹:𝑋𝑌)
2 paste.6 . . . . . . 7 (𝜑 → (𝐴𝐵) = 𝑋)
32ineq2d 4188 . . . . . 6 (𝜑 → ((𝐹𝑦) ∩ (𝐴𝐵)) = ((𝐹𝑦) ∩ 𝑋))
41ffund 6517 . . . . . . . 8 (𝜑 → Fun 𝐹)
5 respreima 6835 . . . . . . . . 9 (Fun 𝐹 → ((𝐹𝐴) “ 𝑦) = ((𝐹𝑦) ∩ 𝐴))
6 respreima 6835 . . . . . . . . 9 (Fun 𝐹 → ((𝐹𝐵) “ 𝑦) = ((𝐹𝑦) ∩ 𝐵))
75, 6uneq12d 4139 . . . . . . . 8 (Fun 𝐹 → (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)) = (((𝐹𝑦) ∩ 𝐴) ∪ ((𝐹𝑦) ∩ 𝐵)))
84, 7syl 17 . . . . . . 7 (𝜑 → (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)) = (((𝐹𝑦) ∩ 𝐴) ∪ ((𝐹𝑦) ∩ 𝐵)))
9 indi 4249 . . . . . . 7 ((𝐹𝑦) ∩ (𝐴𝐵)) = (((𝐹𝑦) ∩ 𝐴) ∪ ((𝐹𝑦) ∩ 𝐵))
108, 9syl6reqr 2875 . . . . . 6 (𝜑 → ((𝐹𝑦) ∩ (𝐴𝐵)) = (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)))
11 imassrn 5939 . . . . . . . . 9 (𝐹𝑦) ⊆ ran 𝐹
12 dfdm4 5763 . . . . . . . . . 10 dom 𝐹 = ran 𝐹
13 fdm 6521 . . . . . . . . . 10 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
1412, 13syl5eqr 2870 . . . . . . . . 9 (𝐹:𝑋𝑌 → ran 𝐹 = 𝑋)
1511, 14sseqtrid 4018 . . . . . . . 8 (𝐹:𝑋𝑌 → (𝐹𝑦) ⊆ 𝑋)
161, 15syl 17 . . . . . . 7 (𝜑 → (𝐹𝑦) ⊆ 𝑋)
17 df-ss 3951 . . . . . . 7 ((𝐹𝑦) ⊆ 𝑋 ↔ ((𝐹𝑦) ∩ 𝑋) = (𝐹𝑦))
1816, 17sylib 220 . . . . . 6 (𝜑 → ((𝐹𝑦) ∩ 𝑋) = (𝐹𝑦))
193, 10, 183eqtr3rd 2865 . . . . 5 (𝜑 → (𝐹𝑦) = (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)))
2019adantr 483 . . . 4 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → (𝐹𝑦) = (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)))
21 paste.4 . . . . . 6 (𝜑𝐴 ∈ (Clsd‘𝐽))
22 paste.8 . . . . . . 7 (𝜑 → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾))
23 cnclima 21875 . . . . . . 7 (((𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐴)))
2422, 23sylan 582 . . . . . 6 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐴)))
25 restcldr 21781 . . . . . 6 ((𝐴 ∈ (Clsd‘𝐽) ∧ ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐴))) → ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘𝐽))
2621, 24, 25syl2an2r 683 . . . . 5 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘𝐽))
27 paste.5 . . . . . 6 (𝜑𝐵 ∈ (Clsd‘𝐽))
28 paste.9 . . . . . . 7 (𝜑 → (𝐹𝐵) ∈ ((𝐽t 𝐵) Cn 𝐾))
29 cnclima 21875 . . . . . . 7 (((𝐹𝐵) ∈ ((𝐽t 𝐵) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐵)))
3028, 29sylan 582 . . . . . 6 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐵)))
31 restcldr 21781 . . . . . 6 ((𝐵 ∈ (Clsd‘𝐽) ∧ ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐵))) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
3227, 30, 31syl2an2r 683 . . . . 5 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
33 uncld 21648 . . . . 5 ((((𝐹𝐴) “ 𝑦) ∈ (Clsd‘𝐽) ∧ ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘𝐽)) → (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
3426, 32, 33syl2anc 586 . . . 4 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
3520, 34eqeltrd 2913 . . 3 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → (𝐹𝑦) ∈ (Clsd‘𝐽))
3635ralrimiva 3182 . 2 (𝜑 → ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))
37 cldrcl 21633 . . . 4 (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
3821, 37syl 17 . . 3 (𝜑𝐽 ∈ Top)
39 cntop2 21848 . . . 4 ((𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾) → 𝐾 ∈ Top)
4022, 39syl 17 . . 3 (𝜑𝐾 ∈ Top)
41 paste.1 . . . . 5 𝑋 = 𝐽
4241toptopon 21524 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
43 paste.2 . . . . 5 𝑌 = 𝐾
4443toptopon 21524 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
45 iscncl 21876 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))
4642, 44, 45syl2anb 599 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))
4738, 40, 46syl2anc 586 . 2 (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))
481, 36, 47mpbir2and 711 1 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138   ∪ cun 3933   ∩ cin 3934   ⊆ wss 3935  ∪ cuni 4837  ◡ccnv 5553  dom cdm 5554  ran crn 5555   ↾ cres 5556   “ cima 5557  Fun wfun 6348  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155   ↾t crest 16693  Topctop 21500  TopOnctopon 21517  Clsdccld 21623   Cn ccn 21831 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-iin 4921  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-oadd 8105  df-er 8288  df-map 8407  df-en 8509  df-fin 8512  df-fi 8874  df-rest 16695  df-topgen 16716  df-top 21501  df-topon 21518  df-bases 21553  df-cld 21626  df-cn 21834 This theorem is referenced by:  cnmpopc  23531  cvmliftlem10  32541
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