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Theorem paste 23318
Description: Pasting lemma. If 𝐴 and 𝐵 are closed sets in 𝑋 with 𝐴𝐵 = 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 4228 . . . . . 6 (𝜑 → ((𝐹𝑦) ∩ (𝐴𝐵)) = ((𝐹𝑦) ∩ 𝑋))
4 indi 4290 . . . . . . 7 ((𝐹𝑦) ∩ (𝐴𝐵)) = (((𝐹𝑦) ∩ 𝐴) ∪ ((𝐹𝑦) ∩ 𝐵))
51ffund 6741 . . . . . . . 8 (𝜑 → Fun 𝐹)
6 respreima 7086 . . . . . . . . 9 (Fun 𝐹 → ((𝐹𝐴) “ 𝑦) = ((𝐹𝑦) ∩ 𝐴))
7 respreima 7086 . . . . . . . . 9 (Fun 𝐹 → ((𝐹𝐵) “ 𝑦) = ((𝐹𝑦) ∩ 𝐵))
86, 7uneq12d 4179 . . . . . . . 8 (Fun 𝐹 → (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)) = (((𝐹𝑦) ∩ 𝐴) ∪ ((𝐹𝑦) ∩ 𝐵)))
95, 8syl 17 . . . . . . 7 (𝜑 → (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)) = (((𝐹𝑦) ∩ 𝐴) ∪ ((𝐹𝑦) ∩ 𝐵)))
104, 9eqtr4id 2794 . . . . . 6 (𝜑 → ((𝐹𝑦) ∩ (𝐴𝐵)) = (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)))
11 imassrn 6091 . . . . . . . . 9 (𝐹𝑦) ⊆ ran 𝐹
12 dfdm4 5909 . . . . . . . . . 10 dom 𝐹 = ran 𝐹
13 fdm 6746 . . . . . . . . . 10 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
1412, 13eqtr3id 2789 . . . . . . . . 9 (𝐹:𝑋𝑌 → ran 𝐹 = 𝑋)
1511, 14sseqtrid 4048 . . . . . . . 8 (𝐹:𝑋𝑌 → (𝐹𝑦) ⊆ 𝑋)
161, 15syl 17 . . . . . . 7 (𝜑 → (𝐹𝑦) ⊆ 𝑋)
17 dfss2 3981 . . . . . . 7 ((𝐹𝑦) ⊆ 𝑋 ↔ ((𝐹𝑦) ∩ 𝑋) = (𝐹𝑦))
1816, 17sylib 218 . . . . . 6 (𝜑 → ((𝐹𝑦) ∩ 𝑋) = (𝐹𝑦))
193, 10, 183eqtr3rd 2784 . . . . 5 (𝜑 → (𝐹𝑦) = (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)))
2019adantr 480 . . . 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 23292 . . . . . . 7 (((𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐴)))
2422, 23sylan 580 . . . . . 6 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐴)))
25 restcldr 23198 . . . . . 6 ((𝐴 ∈ (Clsd‘𝐽) ∧ ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐴))) → ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘𝐽))
2621, 24, 25syl2an2r 685 . . . . 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 23292 . . . . . . 7 (((𝐹𝐵) ∈ ((𝐽t 𝐵) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐵)))
3028, 29sylan 580 . . . . . 6 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐵)))
31 restcldr 23198 . . . . . 6 ((𝐵 ∈ (Clsd‘𝐽) ∧ ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐵))) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
3227, 30, 31syl2an2r 685 . . . . 5 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
33 uncld 23065 . . . . 5 ((((𝐹𝐴) “ 𝑦) ∈ (Clsd‘𝐽) ∧ ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘𝐽)) → (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
3426, 32, 33syl2anc 584 . . . 4 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
3520, 34eqeltrd 2839 . . 3 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → (𝐹𝑦) ∈ (Clsd‘𝐽))
3635ralrimiva 3144 . 2 (𝜑 → ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))
37 cldrcl 23050 . . . 4 (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
3821, 37syl 17 . . 3 (𝜑𝐽 ∈ Top)
39 cntop2 23265 . . . 4 ((𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾) → 𝐾 ∈ Top)
4022, 39syl 17 . . 3 (𝜑𝐾 ∈ Top)
41 paste.1 . . . . 5 𝑋 = 𝐽
4241toptopon 22939 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
43 paste.2 . . . . 5 𝑌 = 𝐾
4443toptopon 22939 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
45 iscncl 23293 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))
4642, 44, 45syl2anb 598 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))
4738, 40, 46syl2anc 584 . 2 (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))
481, 36, 47mpbir2and 713 1 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  cun 3961  cin 3962  wss 3963   cuni 4912  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431  t crest 17467  Topctop 22915  TopOnctopon 22932  Clsdccld 23040   Cn ccn 23248
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-map 8867  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-cn 23251
This theorem is referenced by:  cnmpopc  24969  cvmliftlem10  35279
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