Theorem paste 23316

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 4235	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∩ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝑦) ∩ 𝑋))
4		indi 4297	. . . . . . 7 ⊢ ((◡𝐹 “ 𝑦) ∩ (𝐴 ∪ 𝐵)) = (((◡𝐹 “ 𝑦) ∩ 𝐴) ∪ ((◡𝐹 “ 𝑦) ∩ 𝐵))
5	1	ffund 6750	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
6		respreima 7097	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐴) “ 𝑦) = ((◡𝐹 “ 𝑦) ∩ 𝐴))
7		respreima 7097	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐵) “ 𝑦) = ((◡𝐹 “ 𝑦) ∩ 𝐵))
8	6, 7	uneq12d 4186	. . . . . . . 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 2793	. . . . . 6 ⊢ (𝜑 → ((◡𝐹 “ 𝑦) ∩ (𝐴 ∪ 𝐵)) = ((◡(𝐹 ↾ 𝐴) “ 𝑦) ∪ (◡(𝐹 ↾ 𝐵) “ 𝑦)))
11		imassrn 6099	. . . . . . . . 9 ⊢ (◡𝐹 “ 𝑦) ⊆ ran ◡𝐹
12		dfdm4 5919	. . . . . . . . . 10 ⊢ dom 𝐹 = ran ◡𝐹
13		fdm 6755	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
14	12, 13	eqtr3id 2788	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶𝑌 → ran ◡𝐹 = 𝑋)
15	11, 14	sseqtrid 4055	. . . . . . . 8 ⊢ (𝐹:𝑋⟶𝑌 → (◡𝐹 “ 𝑦) ⊆ 𝑋)
16	1, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → (◡𝐹 “ 𝑦) ⊆ 𝑋)
17		dfss2 3988	. . . . . . 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 2783	. . . . 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 23290	. . . . . . 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 23196	. . . . . 6 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐴))) → (◡(𝐹 ↾ 𝐴) “ 𝑦) ∈ (Clsd‘𝐽))
26	21, 24, 25	syl2an2r 684	. . . . 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 23290	. . . . . . 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 23196	. . . . . 6 ⊢ ((𝐵 ∈ (Clsd‘𝐽) ∧ (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘(𝐽 ↾t 𝐵))) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
32	27, 30, 31	syl2an2r 684	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡(𝐹 ↾ 𝐵) “ 𝑦) ∈ (Clsd‘𝐽))
33		uncld 23063	. . . . 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 2838	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
36	35	ralrimiva 3148	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (Clsd‘𝐾)(◡𝐹 “ 𝑦) ∈ (Clsd‘𝐽))
37		cldrcl 23048	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
38	21, 37	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
39		cntop2 23263	. . . 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 22937	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
43		paste.2	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
44	43	toptopon 22937	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
45		iscncl 23291	. . . 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 712	1 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2103  ∀wral 3063   ∪ cun 3968   ∩ cin 3969   ⊆ wss 3970  ∪ cuni 4931  ^◡ccnv 5698  dom cdm 5699  ran crn 5700   ↾ cres 5701   “ cima 5702  Fun wfun 6566  ⟶wf 6568  ‘cfv 6572  (class class class)co 7445   ↾_t crest 17475  Topctop 22913  TopOnctopon 22930  Clsdccld 23038   Cn ccn 23246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-iin 5022  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-map 8882  df-en 9000  df-fin 9003  df-fi 9476  df-rest 17477  df-topgen 17498  df-top 22914  df-topon 22931  df-bases 22967  df-cld 23041  df-cn 23249

This theorem is referenced by:  cnmpopc  24967  cvmliftlem10  35254