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Theorem paste 21470
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 4042 . . . . . 6 (𝜑 → ((𝐹𝑦) ∩ (𝐴𝐵)) = ((𝐹𝑦) ∩ 𝑋))
41ffund 6283 . . . . . . . 8 (𝜑 → Fun 𝐹)
5 respreima 6594 . . . . . . . . 9 (Fun 𝐹 → ((𝐹𝐴) “ 𝑦) = ((𝐹𝑦) ∩ 𝐴))
6 respreima 6594 . . . . . . . . 9 (Fun 𝐹 → ((𝐹𝐵) “ 𝑦) = ((𝐹𝑦) ∩ 𝐵))
75, 6uneq12d 3996 . . . . . . . 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 4104 . . . . . . 7 ((𝐹𝑦) ∩ (𝐴𝐵)) = (((𝐹𝑦) ∩ 𝐴) ∪ ((𝐹𝑦) ∩ 𝐵))
108, 9syl6reqr 2881 . . . . . 6 (𝜑 → ((𝐹𝑦) ∩ (𝐴𝐵)) = (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)))
11 imassrn 5719 . . . . . . . . 9 (𝐹𝑦) ⊆ ran 𝐹
12 dfdm4 5549 . . . . . . . . . 10 dom 𝐹 = ran 𝐹
13 fdm 6287 . . . . . . . . . 10 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
1412, 13syl5eqr 2876 . . . . . . . . 9 (𝐹:𝑋𝑌 → ran 𝐹 = 𝑋)
1511, 14syl5sseq 3879 . . . . . . . 8 (𝐹:𝑋𝑌 → (𝐹𝑦) ⊆ 𝑋)
161, 15syl 17 . . . . . . 7 (𝜑 → (𝐹𝑦) ⊆ 𝑋)
17 df-ss 3813 . . . . . . 7 ((𝐹𝑦) ⊆ 𝑋 ↔ ((𝐹𝑦) ∩ 𝑋) = (𝐹𝑦))
1816, 17sylib 210 . . . . . 6 (𝜑 → ((𝐹𝑦) ∩ 𝑋) = (𝐹𝑦))
193, 10, 183eqtr3rd 2871 . . . . 5 (𝜑 → (𝐹𝑦) = (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)))
2019adantr 474 . . . 4 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → (𝐹𝑦) = (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)))
21 paste.4 . . . . . . 7 (𝜑𝐴 ∈ (Clsd‘𝐽))
2221adantr 474 . . . . . 6 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → 𝐴 ∈ (Clsd‘𝐽))
23 paste.8 . . . . . . 7 (𝜑 → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾))
24 cnclima 21444 . . . . . . 7 (((𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐴)))
2523, 24sylan 577 . . . . . 6 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐴)))
26 restcldr 21350 . . . . . 6 ((𝐴 ∈ (Clsd‘𝐽) ∧ ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐴))) → ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘𝐽))
2722, 25, 26syl2anc 581 . . . . 5 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐴) “ 𝑦) ∈ (Clsd‘𝐽))
28 paste.5 . . . . . . 7 (𝜑𝐵 ∈ (Clsd‘𝐽))
2928adantr 474 . . . . . 6 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → 𝐵 ∈ (Clsd‘𝐽))
30 paste.9 . . . . . . 7 (𝜑 → (𝐹𝐵) ∈ ((𝐽t 𝐵) Cn 𝐾))
31 cnclima 21444 . . . . . . 7 (((𝐹𝐵) ∈ ((𝐽t 𝐵) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐵)))
3230, 31sylan 577 . . . . . 6 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐵)))
33 restcldr 21350 . . . . . 6 ((𝐵 ∈ (Clsd‘𝐽) ∧ ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘(𝐽t 𝐵))) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
3429, 32, 33syl2anc 581 . . . . 5 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
35 uncld 21217 . . . . 5 ((((𝐹𝐴) “ 𝑦) ∈ (Clsd‘𝐽) ∧ ((𝐹𝐵) “ 𝑦) ∈ (Clsd‘𝐽)) → (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
3627, 34, 35syl2anc 581 . . . 4 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → (((𝐹𝐴) “ 𝑦) ∪ ((𝐹𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
3720, 36eqeltrd 2907 . . 3 ((𝜑𝑦 ∈ (Clsd‘𝐾)) → (𝐹𝑦) ∈ (Clsd‘𝐽))
3837ralrimiva 3176 . 2 (𝜑 → ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))
39 cldrcl 21202 . . . 4 (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
4021, 39syl 17 . . 3 (𝜑𝐽 ∈ Top)
41 cntop2 21417 . . . 4 ((𝐹𝐴) ∈ ((𝐽t 𝐴) Cn 𝐾) → 𝐾 ∈ Top)
4223, 41syl 17 . . 3 (𝜑𝐾 ∈ Top)
43 paste.1 . . . . 5 𝑋 = 𝐽
4443toptopon 21093 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
45 paste.2 . . . . 5 𝑌 = 𝐾
4645toptopon 21093 . . . 4 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
47 iscncl 21445 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))
4844, 46, 47syl2anb 593 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))
4940, 42, 48syl2anc 581 . 2 (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))
501, 38, 49mpbir2and 706 1 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wral 3118  cun 3797  cin 3798  wss 3799   cuni 4659  ccnv 5342  dom cdm 5343  ran crn 5344  cres 5345  cima 5346  Fun wfun 6118  wf 6120  cfv 6124  (class class class)co 6906  t crest 16435  Topctop 21069  TopOnctopon 21086  Clsdccld 21192   Cn ccn 21400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-iin 4744  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-oadd 7831  df-er 8010  df-map 8125  df-en 8224  df-fin 8227  df-fi 8587  df-rest 16437  df-topgen 16458  df-top 21070  df-topon 21087  df-bases 21122  df-cld 21195  df-cn 21403
This theorem is referenced by:  cnmpt2pc  23098  cvmliftlem10  31823
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