Theorem hauseqlcld 23592

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 2735	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
3		eqid 2735	. . . . . . . . . . 11 ⊢ ∪ 𝐾 = ∪ 𝐾
4	2, 3	cnf 23192	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
5	1, 4	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	5	ffvelcdmda 7029	. . . . . . . 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 6846	. . . . . . . . 9 ⊢ (𝐺‘𝑏) ∈ V
9	8	ideq 5800	. . . . . . . 8 ⊢ ((𝐹‘𝑏) I (𝐺‘𝑏) ↔ (𝐹‘𝑏) = (𝐺‘𝑏))
10		df-br 5098	. . . . . . . 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 5945	. . . . . . 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 6833	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
15		fveq2 6833	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝐺‘𝑎) = (𝐺‘𝑏))
16	14, 15	opeq12d 4836	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ = ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩)
17		eqid 2735	. . . . . . . . 9 ⊢ (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) = (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
18		opex 5411	. . . . . . . . 9 ⊢ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ V
19	16, 17, 18	fvmpt 6940	. . . . . . . 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 2820	. . . . . 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 6662	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ∪ 𝐽)
25		hauseqlcld.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
26	2, 3	cnf 23192	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺:∪ 𝐽⟶∪ 𝐾)
27	25, 26	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:∪ 𝐽⟶∪ 𝐾)
28	27	ffnd 6662	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ∪ 𝐽)
29		fndmin 6990	. . . . . . 7 ⊢ ((𝐹 Fn ∪ 𝐽 ∧ 𝐺 Fn ∪ 𝐽) → dom (𝐹 ∩ 𝐺) = {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)})
30	24, 28, 29	syl2anc 585	. . . . . 6 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) = {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)})
31	30	eleq2d 2821	. . . . 5 ⊢ (𝜑 → (𝑏 ∈ dom (𝐹 ∩ 𝐺) ↔ 𝑏 ∈ {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)}))
32		rabid 3419	. . . . 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 5411	. . . . . 6 ⊢ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ V
35	34, 17	fnmpti 6634	. . . . 5 ⊢ (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) Fn ∪ 𝐽
36		elpreima 7003	. . . . 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 2733	. 2 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) = (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾)))
40	2, 17	txcnmpt 23570	. . . 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 585	. . 3 ⊢ (𝜑 → (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾)))
42		hauseqlcld.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Haus)
43	3	hausdiag 23591	. . . . 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 23214	. . 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 585	. 2 ⊢ (𝜑 → (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾)) ∈ (Clsd‘𝐽))
48	39, 47	eqeltrd 2835	1 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3398   ∩ cin 3899  ⟨cop 4585  ∪ cuni 4862   class class class wbr 5097   ↦ cmpt 5178   I cid 5517  ^◡ccnv 5622  dom cdm 5623   ↾ cres 5625   “ cima 5626   Fn wfn 6486  ⟶wf 6487  ‘cfv 6491  (class class class)co 7358  Topctop 22839  Clsdccld 22962   Cn ccn 23170  Hauscha 23254   ×_t ctx 23506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-fv 6499  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8767  df-topgen 17365  df-top 22840  df-topon 22857  df-bases 22892  df-cld 22965  df-cn 23173  df-haus 23261  df-tx 23508

This theorem is referenced by:  hauseqcn  34034  hausgraph  43484