Theorem hauseqcn 34061

Description: In a Hausdorff topology, two continuous functions which agree on a dense set agree everywhere. (Contributed by Thierry Arnoux, 28-Dec-2017.)

Hypotheses

Ref	Expression
hauseqcn.x	⊢ 𝑋 = ∪ 𝐽
hauseqcn.k	⊢ (𝜑 → 𝐾 ∈ Haus)
hauseqcn.f	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
hauseqcn.g	⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
hauseqcn.e	⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
hauseqcn.a	⊢ (𝜑 → 𝐴 ⊆ 𝑋)
hauseqcn.c	⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)

Assertion

Ref	Expression
hauseqcn	⊢ (𝜑 → 𝐹 = 𝐺)

Proof of Theorem hauseqcn

Step	Hyp	Ref	Expression
1		hauseqcn.x	. . 3 ⊢ 𝑋 = ∪ 𝐽
2		hauseqcn.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
3		cntop1 23218	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
4	2, 3	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
5		dmin 5861	. . . . . 6 ⊢ dom (𝐹 ∩ 𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺)
6		eqid 2737	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
7		eqid 2737	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
8	6, 7	cnf 23224	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
9		fdm 6672	. . . . . . . . 9 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → dom 𝐹 = ∪ 𝐽)
10	2, 8, 9	3syl 18	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = ∪ 𝐽)
11		hauseqcn.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
12	6, 7	cnf 23224	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺:∪ 𝐽⟶∪ 𝐾)
13		fdm 6672	. . . . . . . . 9 ⊢ (𝐺:∪ 𝐽⟶∪ 𝐾 → dom 𝐺 = ∪ 𝐽)
14	11, 12, 13	3syl 18	. . . . . . . 8 ⊢ (𝜑 → dom 𝐺 = ∪ 𝐽)
15	10, 14	ineq12d 4162	. . . . . . 7 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (∪ 𝐽 ∩ ∪ 𝐽))
16		inidm 4168	. . . . . . 7 ⊢ (∪ 𝐽 ∩ ∪ 𝐽) = ∪ 𝐽
17	15, 16	eqtrdi 2788	. . . . . 6 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = ∪ 𝐽)
18	5, 17	sseqtrid 3965	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ⊆ ∪ 𝐽)
19		hauseqcn.e	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
20		ffn 6663	. . . . . . . 8 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → 𝐹 Fn ∪ 𝐽)
21	2, 8, 20	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ∪ 𝐽)
22		ffn 6663	. . . . . . . 8 ⊢ (𝐺:∪ 𝐽⟶∪ 𝐾 → 𝐺 Fn ∪ 𝐽)
23	11, 12, 22	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ∪ 𝐽)
24		hauseqcn.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
25	24, 1	sseqtrdi 3963	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
26		fnreseql 6995	. . . . . . 7 ⊢ ((𝐹 Fn ∪ 𝐽 ∧ 𝐺 Fn ∪ 𝐽 ∧ 𝐴 ⊆ ∪ 𝐽) → ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ↔ 𝐴 ⊆ dom (𝐹 ∩ 𝐺)))
27	21, 23, 25, 26	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ↔ 𝐴 ⊆ dom (𝐹 ∩ 𝐺)))
28	19, 27	mpbid 232	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ dom (𝐹 ∩ 𝐺))
29	6	clsss 23032	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ dom (𝐹 ∩ 𝐺) ⊆ ∪ 𝐽 ∧ 𝐴 ⊆ dom (𝐹 ∩ 𝐺)) → ((cls‘𝐽)‘𝐴) ⊆ ((cls‘𝐽)‘dom (𝐹 ∩ 𝐺)))
30	4, 18, 28, 29	syl3anc 1374	. . . 4 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) ⊆ ((cls‘𝐽)‘dom (𝐹 ∩ 𝐺)))
31		hauseqcn.c	. . . 4 ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
32		hauseqcn.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Haus)
33	32, 2, 11	hauseqlcld 23624	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ∈ (Clsd‘𝐽))
34		cldcls 23020	. . . . 5 ⊢ (dom (𝐹 ∩ 𝐺) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘dom (𝐹 ∩ 𝐺)) = dom (𝐹 ∩ 𝐺))
35	33, 34	syl 17	. . . 4 ⊢ (𝜑 → ((cls‘𝐽)‘dom (𝐹 ∩ 𝐺)) = dom (𝐹 ∩ 𝐺))
36	30, 31, 35	3sstr3d 3977	. . 3 ⊢ (𝜑 → 𝑋 ⊆ dom (𝐹 ∩ 𝐺))
37	1, 36	eqsstrrid 3962	. 2 ⊢ (𝜑 → ∪ 𝐽 ⊆ dom (𝐹 ∩ 𝐺))
38		fneqeql2 6994	. . 3 ⊢ ((𝐹 Fn ∪ 𝐽 ∧ 𝐺 Fn ∪ 𝐽) → (𝐹 = 𝐺 ↔ ∪ 𝐽 ⊆ dom (𝐹 ∩ 𝐺)))
39	21, 23, 38	syl2anc 585	. 2 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ ∪ 𝐽 ⊆ dom (𝐹 ∩ 𝐺)))
40	37, 39	mpbird 257	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114   ∩ cin 3889   ⊆ wss 3890  ∪ cuni 4851  dom cdm 5625   ↾ cres 5627   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  Topctop 22871  Clsdccld 22994  clsccl 22996   Cn ccn 23202  Hauscha 23286

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 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-map 8769  df-topgen 17400  df-top 22872  df-topon 22889  df-bases 22924  df-cld 22997  df-cls 22999  df-cn 23205  df-haus 23293  df-tx 23540

This theorem is referenced by:  rrhre  34184