Theorem hauseqlcld 23540

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 2730	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
3		eqid 2730	. . . . . . . . . . 11 ⊢ ∪ 𝐾 = ∪ 𝐾
4	2, 3	cnf 23140	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
5	1, 4	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	5	ffvelcdmda 7059	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → (𝐹‘𝑏) ∈ ∪ 𝐾)
7	6	biantrurd 532	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → (⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ I ↔ ((𝐹‘𝑏) ∈ ∪ 𝐾 ∧ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ I )))
8		fvex 6874	. . . . . . . . 9 ⊢ (𝐺‘𝑏) ∈ V
9	8	ideq 5819	. . . . . . . 8 ⊢ ((𝐹‘𝑏) I (𝐺‘𝑏) ↔ (𝐹‘𝑏) = (𝐺‘𝑏))
10		df-br 5111	. . . . . . . 8 ⊢ ((𝐹‘𝑏) I (𝐺‘𝑏) ↔ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ I )
11	9, 10	bitr3i 277	. . . . . . 7 ⊢ ((𝐹‘𝑏) = (𝐺‘𝑏) ↔ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ I )
12	8	opelresi 5961	. . . . . . 7 ⊢ (⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ ( I ↾ ∪ 𝐾) ↔ ((𝐹‘𝑏) ∈ ∪ 𝐾 ∧ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ I ))
13	7, 11, 12	3bitr4g 314	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → ((𝐹‘𝑏) = (𝐺‘𝑏) ↔ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ ( I ↾ ∪ 𝐾)))
14		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
15		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝐺‘𝑎) = (𝐺‘𝑏))
16	14, 15	opeq12d 4848	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ = ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩)
17		eqid 2730	. . . . . . . . 9 ⊢ (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) = (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
18		opex 5427	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ V
19	16, 17, 18	fvmpt 6971	. . . . . . . 8 ⊢ (𝑏 ∈ ∪ 𝐽 → ((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) = ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩)
20	19	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → ((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) = ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩)
21	20	eleq1d 2814	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → (((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) ∈ ( I ↾ ∪ 𝐾) ↔ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ ( I ↾ ∪ 𝐾)))
22	13, 21	bitr4d 282	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → ((𝐹‘𝑏) = (𝐺‘𝑏) ↔ ((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) ∈ ( I ↾ ∪ 𝐾)))
23	22	pm5.32da 579	. . . 4 ⊢ (𝜑 → ((𝑏 ∈ ∪ 𝐽 ∧ (𝐹‘𝑏) = (𝐺‘𝑏)) ↔ (𝑏 ∈ ∪ 𝐽 ∧ ((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) ∈ ( I ↾ ∪ 𝐾))))
24	5	ffnd 6692	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ∪ 𝐽)
25		hauseqlcld.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
26	2, 3	cnf 23140	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺:∪ 𝐽⟶∪ 𝐾)
27	25, 26	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:∪ 𝐽⟶∪ 𝐾)
28	27	ffnd 6692	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ∪ 𝐽)
29		fndmin 7020	. . . . . . 7 ⊢ ((𝐹 Fn ∪ 𝐽 ∧ 𝐺 Fn ∪ 𝐽) → dom (𝐹 ∩ 𝐺) = {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)})
30	24, 28, 29	syl2anc 584	. . . . . 6 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) = {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)})
31	30	eleq2d 2815	. . . . 5 ⊢ (𝜑 → (𝑏 ∈ dom (𝐹 ∩ 𝐺) ↔ 𝑏 ∈ {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)}))
32		rabid 3430	. . . . 5 ⊢ (𝑏 ∈ {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)} ↔ (𝑏 ∈ ∪ 𝐽 ∧ (𝐹‘𝑏) = (𝐺‘𝑏)))
33	31, 32	bitrdi 287	. . . 4 ⊢ (𝜑 → (𝑏 ∈ dom (𝐹 ∩ 𝐺) ↔ (𝑏 ∈ ∪ 𝐽 ∧ (𝐹‘𝑏) = (𝐺‘𝑏))))
34		opex 5427	. . . . . 6 ⊢ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ V
35	34, 17	fnmpti 6664	. . . . 5 ⊢ (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) Fn ∪ 𝐽
36		elpreima 7033	. . . . 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 13	. . . 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 311	. . 3 ⊢ (𝜑 → (𝑏 ∈ dom (𝐹 ∩ 𝐺) ↔ 𝑏 ∈ (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾))))
39	38	eqrdv 2728	. 2 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) = (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾)))
40	2, 17	txcnmpt 23518	. . . 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 584	. . 3 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾)))
42		hauseqlcld.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Haus)
43	3	hausdiag 23539	. . . . 5 ⊢ (𝐾 ∈ Haus ↔ (𝐾 ∈ Top ∧ ( I ↾ ∪ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾))))
44	43	simprbi 496	. . . 4 ⊢ (𝐾 ∈ Haus → ( I ↾ ∪ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))
45	42, 44	syl 17	. . 3 ⊢ (𝜑 → ( I ↾ ∪ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))
46		cnclima 23162	. . 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 584	. 2 ⊢ (𝜑 → (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾)) ∈ (Clsd‘𝐽))
48	39, 47	eqeltrd 2829	1 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3408   ∩ cin 3916  ⟨cop 4598  ∪ cuni 4874   class class class wbr 5110   ↦ cmpt 5191   I cid 5535  ^◡ccnv 5640  dom cdm 5641   ↾ cres 5643   “ cima 5644   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Topctop 22787  Clsdccld 22910   Cn ccn 23118  Hauscha 23202   ×_t ctx 23454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-topgen 17413  df-top 22788  df-topon 22805  df-bases 22840  df-cld 22913  df-cn 23121  df-haus 23209  df-tx 23456

This theorem is referenced by:  hauseqcn  33895  hausgraph  43201