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Theorem paste 21296
 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 3953 . . . . . 6 (𝜑 → ((𝐹𝑦) ∩ (𝐴𝐵)) = ((𝐹𝑦) ∩ 𝑋))
4 ffun 6205 . . . . . . . . 9 (𝐹:𝑋𝑌 → Fun 𝐹)
51, 4syl 17 . . . . . . . 8 (𝜑 → Fun 𝐹)
6 respreima 6503 . . . . . . . . 9 (Fun 𝐹 → ((𝐹𝐴) “ 𝑦) = ((𝐹𝑦) ∩ 𝐴))
7 respreima 6503 . . . . . . . . 9 (Fun 𝐹 → ((𝐹𝐵) “ 𝑦) = ((𝐹𝑦) ∩ 𝐵))
86, 7uneq12d 3907 . . . . . . . 8 (Fun 𝐹 → (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)) = (((𝐹𝑦) ∩ 𝐴) ∪ ((𝐹𝑦) ∩ 𝐵)))
95, 8syl 17 . . . . . . 7 (𝜑 → (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)) = (((𝐹𝑦) ∩ 𝐴) ∪ ((𝐹𝑦) ∩ 𝐵)))
10 indi 4012 . . . . . . 7 ((𝐹𝑦) ∩ (𝐴𝐵)) = (((𝐹𝑦) ∩ 𝐴) ∪ ((𝐹𝑦) ∩ 𝐵))
119, 10syl6reqr 2809 . . . . . 6 (𝜑 → ((𝐹𝑦) ∩ (𝐴𝐵)) = (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)))
12 imassrn 5631 . . . . . . . . 9 (𝐹𝑦) ⊆ ran 𝐹
13 dfdm4 5467 . . . . . . . . . 10 dom 𝐹 = ran 𝐹
14 fdm 6208 . . . . . . . . . 10 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
1513, 14syl5eqr 2804 . . . . . . . . 9 (𝐹:𝑋𝑌 → ran 𝐹 = 𝑋)
1612, 15syl5sseq 3790 . . . . . . . 8 (𝐹:𝑋𝑌 → (𝐹𝑦) ⊆ 𝑋)
171, 16syl 17 . . . . . . 7 (𝜑 → (𝐹𝑦) ⊆ 𝑋)
18 df-ss 3725 . . . . . . 7 ((𝐹𝑦) ⊆ 𝑋 ↔ ((𝐹𝑦) ∩ 𝑋) = (𝐹𝑦))
1917, 18sylib 208 . . . . . 6 (𝜑 → ((𝐹𝑦) ∩ 𝑋) = (𝐹𝑦))
203, 11, 193eqtr3rd 2799 . . . . 5 (𝜑 → (𝐹𝑦) = (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)))
2120adantr 472 . . . 4 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → (𝐹𝑦) = (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)))
22 paste.4 . . . . . . 7 (𝜑𝐴 ∈ (Clsd‘𝐽))
2322adantr 472 . . . . . 6 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → 𝐴 ∈ (Clsd‘𝐽))
24 paste.8 . . . . . . 7 (𝜑 → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾))
25 cnclima 21270 . . . . . . 7 (((𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐴)))
2624, 25sylan 489 . . . . . 6 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐴)))
27 restcldr 21176 . . . . . 6 ((𝐴 ∈ (Clsd‘𝐽) ∧ ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐴))) → ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘𝐽))
2823, 26, 27syl2anc 696 . . . . 5 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘𝐽))
29 paste.5 . . . . . . 7 (𝜑𝐵 ∈ (Clsd‘𝐽))
3029adantr 472 . . . . . 6 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → 𝐵 ∈ (Clsd‘𝐽))
31 paste.9 . . . . . . 7 (𝜑 → (𝐹𝐵) ∈ ((𝐽t 𝐵) Cn 𝐾))
32 cnclima 21270 . . . . . . 7 (((𝐹𝐵) ∈ ((𝐽t 𝐵) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐵)))
3331, 32sylan 489 . . . . . 6 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐵)))
34 restcldr 21176 . . . . . 6 ((𝐵 ∈ (Clsd‘𝐽) ∧ ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐵))) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
3530, 33, 34syl2anc 696 . . . . 5 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
36 uncld 21043 . . . . 5 ((((𝐹𝐴) “ 𝑦) ∈ (Clsd‘𝐽) ∧ ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘𝐽)) → (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
3728, 35, 36syl2anc 696 . . . 4 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
3821, 37eqeltrd 2835 . . 3 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → (𝐹𝑦) ∈ (Clsd‘𝐽))
3938ralrimiva 3100 . 2 (𝜑 → ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))
40 cldrcl 21028 . . . 4 (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
4122, 40syl 17 . . 3 (𝜑𝐽 ∈ Top)
42 cntop2 21243 . . . 4 ((𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾) → 𝐾 ∈ Top)
4324, 42syl 17 . . 3 (𝜑𝐾 ∈ Top)
44 paste.1 . . . . 5 𝑋 = 𝐽
4544toptopon 20920 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
46 paste.2 . . . . 5 𝑌 = 𝐾
4746toptopon 20920 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
48 iscncl 21271 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))
4945, 47, 48syl2anb 497 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))
5041, 43, 49syl2anc 696 . 2 (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))
511, 39, 50mpbir2and 995 1 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1628   ∈ wcel 2135  ∀wral 3046   ∪ cun 3709   ∩ cin 3710   ⊆ wss 3711  ∪ cuni 4584  ◡ccnv 5261  dom cdm 5262  ran crn 5263   ↾ cres 5264   “ cima 5265  Fun wfun 6039  ⟶wf 6041  ‘cfv 6045  (class class class)co 6809   ↾t crest 16279  Topctop 20896  TopOnctopon 20913  Clsdccld 21018   Cn ccn 21226 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-iin 4671  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-oadd 7729  df-er 7907  df-map 8021  df-en 8118  df-fin 8121  df-fi 8478  df-rest 16281  df-topgen 16302  df-top 20897  df-topon 20914  df-bases 20948  df-cld 21021  df-cn 21229 This theorem is referenced by:  cnmpt2pc  22924  cvmliftlem10  31579
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