Theorem hauseqlcld 23818

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 2766	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
3		eqid 2766	. . . . . . . . . . 11 ⊢ ∪ 𝐾 = ∪ 𝐾
4	2, 3	cnf 23418	. . . . . . . . . 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 542	. . . . . . 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 5841	. . . . . . . 8 ⊢ ((𝐹‘𝑏) I (𝐺‘𝑏) ↔ (𝐹‘𝑏) = (𝐺‘𝑏))
10		df-br 5113	. . . . . . . 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 5989	. . . . . . 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 4849	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ = ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩)
17		eqid 2766	. . . . . . . . 9 ⊢ (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) = (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
18		opex 5448	. . . . . . . . 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 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → ((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) = ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩)
21	20	eleq1d 2851	. . . . . 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 590	. . . 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 23418	. . . . . . . . 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 596	. . . . . 6 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) = {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)})
31	30	eleq2d 2852	. . . . 5 ⊢ (𝜑 → (𝑏 ∈ dom (𝐹 ∩ 𝐺) ↔ 𝑏 ∈ {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)}))
32		rabid 3440	. . . . 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 5448	. . . . . 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 2764	. 2 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) = (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾)))
40	2, 17	txcnmpt 23796	. . . 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 596	. . 3 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾)))
42		hauseqlcld.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Haus)
43	3	hausdiag 23817	. . . . 5 ⊢ (𝐾 ∈ Haus ↔ (𝐾 ∈ Top ∧ ( I ↾ ∪ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾))))
44	43	simprbi 503	. . . 4 ⊢ (𝐾 ∈ Haus → ( I ↾ ∪ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))
45	42, 44	syl 18	. . 3 ⊢ (𝜑 → ( I ↾ ∪ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))
46		cnclima 23440	. . 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 596	. 2 ⊢ (𝜑 → (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾)) ∈ (Clsd‘𝐽))
48	39, 47	eqeltrd 2866	1 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ∈ (Clsd‘𝐽))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {crab 3419   ∩ cin 3907  ⟨cop 4598  ∪ cuni 4875   class class class wbr 5112   ↦ cmpt 5195   I cid 5558  ^◡ccnv 5663  dom cdm 5664   ↾ cres 5666   “ cima 5667   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416  Topctop 23065  Clsdccld 23188   Cn ccn 23396  Hauscha 23480   ×_t ctx 23732

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5260  ax-nul 5272  ax-pow 5339  ax-pr 5407  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4876  df-iun 4961  df-br 5113  df-opab 5177  df-mpt 5196  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-map 8828  df-topgen 17506  df-top 23066  df-topon 23083  df-bases 23118  df-cld 23191  df-cn 23399  df-haus 23487  df-tx 23734

This theorem is used by:  hauseqcn  34301  hausgraph  43964