Theorem hauseqlcld 23786

Description: In a Hausdorff topology, the equalizer of two continuous functions is closed (thus, two continuous functions which agree on a dense set agree everywhere). (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
hauseqlcld.k	⊢ (𝜑 → 𝐾 ∈ Haus)
hauseqlcld.f	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
hauseqlcld.g	⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))

Assertion

Ref	Expression
hauseqlcld	⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ∈ (Clsd‘𝐽))

Proof of Theorem hauseqlcld

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hauseqlcld.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
2		eqid 2770	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
3		eqid 2770	. . . . . . . . . . 11 ⊢ ∪ 𝐾 = ∪ 𝐾
4	2, 3	cnf 23386	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
5	1, 4	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	5	ffvelcdmda 7083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → (𝐹‘𝑏) ∈ ∪ 𝐾)
7	6	biantrurd 541	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → (⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ I ↔ ((𝐹‘𝑏) ∈ ∪ 𝐾 ∧ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ I )))
8		fvex 6898	. . . . . . . . 9 ⊢ (𝐺‘𝑏) ∈ V
9	8	ideq 5842	. . . . . . . 8 ⊢ ((𝐹‘𝑏) I (𝐺‘𝑏) ↔ (𝐹‘𝑏) = (𝐺‘𝑏))
10		df-br 5115	. . . . . . . 8 ⊢ ((𝐹‘𝑏) I (𝐺‘𝑏) ↔ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ I )
11	9, 10	bitr3i 280	. . . . . . 7 ⊢ ((𝐹‘𝑏) = (𝐺‘𝑏) ↔ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ I )
12	8	opelresi 5990	. . . . . . 7 ⊢ (⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ ( I ↾ ∪ 𝐾) ↔ ((𝐹‘𝑏) ∈ ∪ 𝐾 ∧ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ I ))
13	7, 11, 12	3bitr4g 317	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → ((𝐹‘𝑏) = (𝐺‘𝑏) ↔ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ ( I ↾ ∪ 𝐾)))
14		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
15		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝐺‘𝑎) = (𝐺‘𝑏))
16	14, 15	opeq12d 4851	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ = ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩)
17		eqid 2770	. . . . . . . . 9 ⊢ (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) = (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
18		opex 5449	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ V
19	16, 17, 18	fvmpt 6993	. . . . . . . 8 ⊢ (𝑏 ∈ ∪ 𝐽 → ((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) = ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩)
20	19	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → ((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) = ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩)
21	20	eleq1d 2855	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → (((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) ∈ ( I ↾ ∪ 𝐾) ↔ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ ( I ↾ ∪ 𝐾)))
22	13, 21	bitr4d 285	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → ((𝐹‘𝑏) = (𝐺‘𝑏) ↔ ((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) ∈ ( I ↾ ∪ 𝐾)))
23	22	pm5.32da 589	. . . 4 ⊢ (𝜑 → ((𝑏 ∈ ∪ 𝐽 ∧ (𝐹‘𝑏) = (𝐺‘𝑏)) ↔ (𝑏 ∈ ∪ 𝐽 ∧ ((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) ∈ ( I ↾ ∪ 𝐾))))
24	5	ffnd 6710	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ∪ 𝐽)
25		hauseqlcld.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
26	2, 3	cnf 23386	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺:∪ 𝐽⟶∪ 𝐾)
27	25, 26	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐺:∪ 𝐽⟶∪ 𝐾)
28	27	ffnd 6710	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ∪ 𝐽)
29		fndmin 7044	. . . . . . 7 ⊢ ((𝐹 Fn ∪ 𝐽 ∧ 𝐺 Fn ∪ 𝐽) → dom (𝐹 ∩ 𝐺) = {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)})
30	24, 28, 29	syl2anc 595	. . . . . 6 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) = {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)})
31	30	eleq2d 2856	. . . . 5 ⊢ (𝜑 → (𝑏 ∈ dom (𝐹 ∩ 𝐺) ↔ 𝑏 ∈ {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)}))
32		rabid 3444	. . . . 5 ⊢ (𝑏 ∈ {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)} ↔ (𝑏 ∈ ∪ 𝐽 ∧ (𝐹‘𝑏) = (𝐺‘𝑏)))
33	31, 32	bitrdi 290	. . . 4 ⊢ (𝜑 → (𝑏 ∈ dom (𝐹 ∩ 𝐺) ↔ (𝑏 ∈ ∪ 𝐽 ∧ (𝐹‘𝑏) = (𝐺‘𝑏))))
34		opex 5449	. . . . . 6 ⊢ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ V
35	34, 17	fnmpti 6682	. . . . 5 ⊢ (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) Fn ∪ 𝐽
36		elpreima 7057	. . . . 5 ⊢ ((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) Fn ∪ 𝐽 → (𝑏 ∈ (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾)) ↔ (𝑏 ∈ ∪ 𝐽 ∧ ((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) ∈ ( I ↾ ∪ 𝐾))))
37	35, 36	mp1i 14	. . . 4 ⊢ (𝜑 → (𝑏 ∈ (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾)) ↔ (𝑏 ∈ ∪ 𝐽 ∧ ((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) ∈ ( I ↾ ∪ 𝐾))))
38	23, 33, 37	3bitr4d 314	. . 3 ⊢ (𝜑 → (𝑏 ∈ dom (𝐹 ∩ 𝐺) ↔ 𝑏 ∈ (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾))))
39	38	eqrdv 2768	. 2 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) = (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾)))
40	2, 17	txcnmpt 23764	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐽 Cn 𝐾)) → (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾)))
41	1, 25, 40	syl2anc 595	. . 3 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾)))
42		hauseqlcld.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Haus)
43	3	hausdiag 23785	. . . . 5 ⊢ (𝐾 ∈ Haus ↔ (𝐾 ∈ Top ∧ ( I ↾ ∪ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾))))
44	43	simprbi 502	. . . 4 ⊢ (𝐾 ∈ Haus → ( I ↾ ∪ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))
45	42, 44	syl 18	. . 3 ⊢ (𝜑 → ( I ↾ ∪ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))
46		cnclima 23408	. . 3 ⊢ (((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾)) ∧ ( I ↾ ∪ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾))) → (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾)) ∈ (Clsd‘𝐽))
47	41, 45, 46	syl2anc 595	. 2 ⊢ (𝜑 → (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾)) ∈ (Clsd‘𝐽))
48	39, 47	eqeltrd 2870	1 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2150  {crab 3423   ∩ cin 3912  ⟨cop 4600  ∪ cuni 4877   class class class wbr 5114   ↦ cmpt 5197   I cid 5559  ^◡ccnv 5664  dom cdm 5665   ↾ cres 5667   “ cima 5668   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7414  Topctop 23033  Clsdccld 23156   Cn ccn 23364  Hauscha 23448   ×_t ctx 23700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7989  df-2nd 7990  df-map 8829  df-topgen 17499  df-top 23034  df-topon 23051  df-bases 23086  df-cld 23159  df-cn 23367  df-haus 23455  df-tx 23702

This theorem is referenced by:  hauseqcn  34258  hausgraph  43884