Theorem paste 23548

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	⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))
paste.5	⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽))
paste.6	⊢ (𝜑 → (𝐴 ∪ 𝐵) = 𝑋)
paste.7	⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
paste.8	⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))
paste.9	⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ ((𝐽 ↾t 𝐵) Cn 𝐾))

Assertion

Ref	Expression
paste	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))

Proof of Theorem paste

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		paste.7	. 2 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
2		paste.6	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = 𝑋)
3	2	ineq2d 4166	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∩ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝑦) ∩ 𝑋))
4		indi 4230	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑦) ∩ (𝐴 ∪ 𝐵)) = (((◡𝐹 “ 𝑦) ∩ 𝐴) ∪ ((◡𝐹 “ 𝑦) ∩ 𝐵))
5	1	ffund 6709	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
6		respreima 7060	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐴) “ 𝑦) = ((◡𝐹 “ 𝑦) ∩ 𝐴))
7		respreima 7060	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐵) “ 𝑦) = ((◡𝐹 “ 𝑦) ∩ 𝐵))
8	6, 7	uneq12d 4116	. . . . . . . 8 ⊢ (Fun 𝐹 → ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)) = (((◡𝐹 “ 𝑦) ∩ 𝐴) ∪ ((◡𝐹 “ 𝑦) ∩ 𝐵)))
9	5, 8	syl 18	. . . . . . 7 ⊢ (𝜑 → ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)) = (((◡𝐹 “ 𝑦) ∩ 𝐴) ∪ ((◡𝐹 “ 𝑦) ∩ 𝐵)))
10	4, 9	eqtr4id 2814	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∩ (𝐴 ∪ 𝐵)) = ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)))
11		imassrn 6068	. . . . . . . . 9 ⊢ (◡𝐹 “ 𝑦) ⊆ ran ◡𝐹
12		dfdm4 5880	. . . . . . . . . 10 ⊢ dom 𝐹 = ran ◡𝐹
13		fdm 6714	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
14	12, 13	eqtr3id 2809	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶𝑌 → ran ◡𝐹 = 𝑋)
15	11, 14	sseqtrid 3973	. . . . . . . 8 ⊢ (𝐹:𝑋⟶𝑌 → (◡𝐹 “ 𝑦) ⊆ 𝑋)
16	1, 15	syl 18	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ 𝑦) ⊆ 𝑋)
17		dfss2 3917	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑦) ⊆ 𝑋 ↔ ((◡𝐹 “ 𝑦) ∩ 𝑋) = (◡𝐹 “ 𝑦))
18	16, 17	sylib 221	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∩ 𝑋) = (◡𝐹 “ 𝑦))
19	3, 10, 18	3eqtr3rd 2804	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ 𝑦) = ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)))
20	19	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝑦) = ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)))
21		paste.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))
22		paste.8	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))
23		cnclima 23522	. . . . . . 7 ⊢ (((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
24	22, 23	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
25		restcldr 23428	. . . . . 6 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐴))) → (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘𝐽))
26	21, 24, 25	syl2an2r 698	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘𝐽))
27		paste.5	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽))
28		paste.9	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ ((𝐽 ↾t 𝐵) Cn 𝐾))
29		cnclima 23522	. . . . . . 7 ⊢ (((𝐹 ↾ 𝐵) ∈ ((𝐽 ↾t 𝐵) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐵)))
30	28, 29	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐵)))
31		restcldr 23428	. . . . . 6 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐵))) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
32	27, 30, 31	syl2an2r 698	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
33		uncld 23295	. . . . 5 ⊢ (((◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘𝐽) ∧ (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘𝐽)) → ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
34	26, 32, 33	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
35	20, 34	eqeltrd 2860	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
36	35	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
37		cldrcl 23280	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
38	21, 37	syl 18	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
39		cntop2 23495	. . . 4 ⊢ ((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) → 𝐾 ∈ Top)
40	22, 39	syl 18	. . 3 ⊢ (𝜑 → 𝐾 ∈ Top)
41		paste.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
42	41	toptopon 23171	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
43		paste.2	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
44	43	toptopon 23171	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
45		iscncl 23523	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))))
46	42, 44, 45	syl2anb 610	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))))
47	38, 40, 46	syl2anc 596	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))))
48	1, 36, 47	mpbir2and 726	1 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∪ cuni 4867  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   ↾ cres 5657   “ cima 5658  Fun wfun 6528  ⟶wf 6530  ‘cfv 6534  (class class class)co 7415   ↾_t crest 17527  Topctop 23147  TopOnctopon 23164  Clsdccld 23270   Cn ccn 23478

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7865  df-1st 7988  df-2nd 7989  df-map 8832  df-en 8957  df-fin 8960  df-fi 9385  df-rest 17529  df-topgen 17550  df-top 23148  df-topon 23165  df-bases 23200  df-cld 23273  df-cn 23481

This theorem is used by:  cnmpopc  25185  cvmliftlem10  35903