Theorem paste 20908

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 3776	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∩ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝑦) ∩ 𝑋))
4		ffun 5961	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹)
5	1, 4	syl 17	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
6		respreima 6252	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐴) “ 𝑦) = ((◡𝐹 “ 𝑦) ∩ 𝐴))
7		respreima 6252	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐵) “ 𝑦) = ((◡𝐹 “ 𝑦) ∩ 𝐵))
8	6, 7	uneq12d 3730	. . . . . . . 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		indi 3832	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑦) ∩ (𝐴 ∪ 𝐵)) = (((◡𝐹 “ 𝑦) ∩ 𝐴) ∪ ((◡𝐹 “ 𝑦) ∩ 𝐵))
11	9, 10	syl6reqr 2663	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∩ (𝐴 ∪ 𝐵)) = ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)))
12		imassrn 5396	. . . . . . . . 9 ⊢ (◡𝐹 “ 𝑦) ⊆ ran ◡𝐹
13		dfdm4 5238	. . . . . . . . . 10 ⊢ dom 𝐹 = ran ◡𝐹
14		fdm 5964	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
15	13, 14	syl5eqr 2658	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶𝑌 → ran ◡𝐹 = 𝑋)
16	12, 15	syl5sseq 3616	. . . . . . . 8 ⊢ (𝐹:𝑋⟶𝑌 → (◡𝐹 “ 𝑦) ⊆ 𝑋)
17	1, 16	syl 17	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ 𝑦) ⊆ 𝑋)
18		df-ss 3554	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑦) ⊆ 𝑋 ↔ ((◡𝐹 “ 𝑦) ∩ 𝑋) = (◡𝐹 “ 𝑦))
19	17, 18	sylib 207	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∩ 𝑋) = (◡𝐹 “ 𝑦))
20	3, 11, 19	3eqtr3rd 2653	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ 𝑦) = ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)))
21	20	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝑦) = ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)))
22		paste.4	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))
23	22	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → 𝐴 ∈ (Clsd‘𝐽))
24		paste.8	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))
25		cnclima 20882	. . . . . . 7 ⊢ (((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
26	24, 25	sylan 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐴)))
27		restcldr 20788	. . . . . 6 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐴))) → (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘𝐽))
28	23, 26, 27	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘𝐽))
29		paste.5	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽))
30	29	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → 𝐵 ∈ (Clsd‘𝐽))
31		paste.9	. . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ ((𝐽 ↾t 𝐵) Cn 𝐾))
32		cnclima 20882	. . . . . . 7 ⊢ (((𝐹 ↾ 𝐵) ∈ ((𝐽 ↾t 𝐵) Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐵)))
33	31, 32	sylan 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐵)))
34		restcldr 20788	. . . . . 6 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐵))) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
35	30, 33, 34	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
36		uncld 20655	. . . . 5 ⊢ (((◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘𝐽) ∧ (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘𝐽)) → ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
37	28, 35, 36	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)) ∈ (Clsd‘𝐽))
38	21, 37	eqeltrd 2688	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
39	38	ralrimiva 2949	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
40		cldrcl 20640	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
41	22, 40	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
42		cntop2 20855	. . . 4 ⊢ ((𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) → 𝐾 ∈ Top)
43	24, 42	syl 17	. . 3 ⊢ (𝜑 → 𝐾 ∈ Top)
44		paste.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
45	44	toptopon 20548	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
46		paste.2	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
47	46	toptopon 20548	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
48		iscncl 20883	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))))
49	45, 47, 48	syl2anb 495	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))))
50	41, 43, 49	syl2anc 691	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))))
51	1, 39, 50	mpbir2and 959	1 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∪ cuni 4372  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↾_t crest 15904  Topctop 20517  TopOnctopon 20518  Clsdccld 20630   Cn ccn 20838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-cn 20841

This theorem is referenced by:  cnmpt2pc  22535  cvmliftlem10  30530