Theorem paste 22353

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 4143	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∩ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝑦) ∩ 𝑋))
4		indi 4204	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑦) ∩ (𝐴 ∪ 𝐵)) = (((◡𝐹 “ 𝑦) ∩ 𝐴) ∪ ((◡𝐹 “ 𝑦) ∩ 𝐵))
5	1	ffund 6588	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
6		respreima 6925	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐴) “ 𝑦) = ((◡𝐹 “ 𝑦) ∩ 𝐴))
7		respreima 6925	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐵) “ 𝑦) = ((◡𝐹 “ 𝑦) ∩ 𝐵))
8	6, 7	uneq12d 4094	. . . . . . . 8 ⊢ (Fun 𝐹 → ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)) = (((◡𝐹 “ 𝑦) ∩ 𝐴) ∪ ((◡𝐹 “ 𝑦) ∩ 𝐵)))
9	5, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)) = (((◡𝐹 “ 𝑦) ∩ 𝐴) ∪ ((◡𝐹 “ 𝑦) ∩ 𝐵)))
10	4, 9	eqtr4id 2798	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∩ (𝐴 ∪ 𝐵)) = ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)))
11		imassrn 5969	. . . . . . . . 9 ⊢ (◡𝐹 “ 𝑦) ⊆ ran ◡𝐹
12		dfdm4 5793	. . . . . . . . . 10 ⊢ dom 𝐹 = ran ◡𝐹
13		fdm 6593	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
14	12, 13	eqtr3id 2793	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶𝑌 → ran ◡𝐹 = 𝑋)
15	11, 14	sseqtrid 3969	. . . . . . . 8 ⊢ (𝐹:𝑋⟶𝑌 → (◡𝐹 “ 𝑦) ⊆ 𝑋)
16	1, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ 𝑦) ⊆ 𝑋)
17		df-ss 3900	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑦) ⊆ 𝑋 ↔ ((◡𝐹 “ 𝑦) ∩ 𝑋) = (◡𝐹 “ 𝑦))
18	16, 17	sylib 217	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∩ 𝑋) = (◡𝐹 “ 𝑦))
19	3, 10, 18	3eqtr3rd 2787	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ 𝑦) = ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)))
20	19	adantr 480	. . . 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 22327	. . . . . . 7 ⊢ (((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
24	22, 23	sylan 579	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
25		restcldr 22233	. . . . . 6 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐴))) → (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘𝐽))
26	21, 24, 25	syl2an2r 681	. . . . 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 22327	. . . . . . 7 ⊢ (((𝐹 ↾ 𝐵) ∈ ((𝐽 ↾t 𝐵) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐵)))
30	28, 29	sylan 579	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐵)))
31		restcldr 22233	. . . . . 6 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐵))) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
32	27, 30, 31	syl2an2r 681	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
33		uncld 22100	. . . . 5 ⊢ (((◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘𝐽) ∧ (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘𝐽)) → ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
34	26, 32, 33	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
35	20, 34	eqeltrd 2839	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
36	35	ralrimiva 3107	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
37		cldrcl 22085	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
38	21, 37	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
39		cntop2 22300	. . . 4 ⊢ ((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) → 𝐾 ∈ Top)
40	22, 39	syl 17	. . 3 ⊢ (𝜑 → 𝐾 ∈ Top)
41		paste.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
42	41	toptopon 21974	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
43		paste.2	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
44	43	toptopon 21974	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
45		iscncl 22328	. . . 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 597	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))))
47	38, 40, 46	syl2anc 583	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))))
48	1, 36, 47	mpbir2and 709	1 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∪ cuni 4836  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   ↾ cres 5582   “ cima 5583  Fun wfun 6412  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↾_t crest 17048  Topctop 21950  TopOnctopon 21967  Clsdccld 22075   Cn ccn 22283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-map 8575  df-en 8692  df-fin 8695  df-fi 9100  df-rest 17050  df-topgen 17071  df-top 21951  df-topon 21968  df-bases 22004  df-cld 22078  df-cn 22286

This theorem is referenced by:  cnmpopc  23997  cvmliftlem10  33156