Theorem paste 23240

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 4172	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∩ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝑦) ∩ 𝑋))
4		indi 4236	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑦) ∩ (𝐴 ∪ 𝐵)) = (((◡𝐹 “ 𝑦) ∩ 𝐴) ∪ ((◡𝐹 “ 𝑦) ∩ 𝐵))
5	1	ffund 6666	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
6		respreima 7011	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐴) “ 𝑦) = ((◡𝐹 “ 𝑦) ∩ 𝐴))
7		respreima 7011	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐵) “ 𝑦) = ((◡𝐹 “ 𝑦) ∩ 𝐵))
8	6, 7	uneq12d 4121	. . . . . . . 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 2790	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∩ (𝐴 ∪ 𝐵)) = ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)))
11		imassrn 6030	. . . . . . . . 9 ⊢ (◡𝐹 “ 𝑦) ⊆ ran ◡𝐹
12		dfdm4 5844	. . . . . . . . . 10 ⊢ dom 𝐹 = ran ◡𝐹
13		fdm 6671	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
14	12, 13	eqtr3id 2785	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶𝑌 → ran ◡𝐹 = 𝑋)
15	11, 14	sseqtrid 3976	. . . . . . . 8 ⊢ (𝐹:𝑋⟶𝑌 → (◡𝐹 “ 𝑦) ⊆ 𝑋)
16	1, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ 𝑦) ⊆ 𝑋)
17		dfss2 3919	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑦) ⊆ 𝑋 ↔ ((◡𝐹 “ 𝑦) ∩ 𝑋) = (◡𝐹 “ 𝑦))
18	16, 17	sylib 218	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∩ 𝑋) = (◡𝐹 “ 𝑦))
19	3, 10, 18	3eqtr3rd 2780	. . . . 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 23214	. . . . . . 7 ⊢ (((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
24	22, 23	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
25		restcldr 23120	. . . . . 6 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐴))) → (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘𝐽))
26	21, 24, 25	syl2an2r 685	. . . . 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 23214	. . . . . . 7 ⊢ (((𝐹 ↾ 𝐵) ∈ ((𝐽 ↾t 𝐵) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐵)))
30	28, 29	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐵)))
31		restcldr 23120	. . . . . 6 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐵))) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
32	27, 30, 31	syl2an2r 685	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
33		uncld 22987	. . . . 5 ⊢ (((◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘𝐽) ∧ (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘𝐽)) → ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
34	26, 32, 33	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
35	20, 34	eqeltrd 2836	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
36	35	ralrimiva 3128	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
37		cldrcl 22972	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
38	21, 37	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
39		cntop2 23187	. . . 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 22863	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
43		paste.2	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
44	43	toptopon 22863	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
45		iscncl 23215	. . . 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 598	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))))
47	38, 40, 46	syl2anc 584	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))))
48	1, 36, 47	mpbir2and 713	1 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ∪ cun 3899   ∩ cin 3900   ⊆ wss 3901  ∪ cuni 4863  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   ↾ cres 5626   “ cima 5627  Fun wfun 6486  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ↾_t crest 17342  Topctop 22839  TopOnctopon 22856  Clsdccld 22962   Cn ccn 23170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-map 8767  df-en 8886  df-fin 8889  df-fi 9316  df-rest 17344  df-topgen 17365  df-top 22840  df-topon 22857  df-bases 22892  df-cld 22965  df-cn 23173

This theorem is referenced by:  cnmpopc  24880  cvmliftlem10  35490