Theorem hauseqcn 31750

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 22299	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
4	2, 3	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
5		dmin 5809	. . . . . 6 ⊢ dom (𝐹 ∩ 𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺)
6		eqid 2738	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
7		eqid 2738	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
8	6, 7	cnf 22305	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
9		fdm 6593	. . . . . . . . 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 22305	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺:∪ 𝐽⟶∪ 𝐾)
13		fdm 6593	. . . . . . . . 9 ⊢ (𝐺:∪ 𝐽⟶∪ 𝐾 → dom 𝐺 = ∪ 𝐽)
14	11, 12, 13	3syl 18	. . . . . . . 8 ⊢ (𝜑 → dom 𝐺 = ∪ 𝐽)
15	10, 14	ineq12d 4144	. . . . . . 7 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (∪ 𝐽 ∩ ∪ 𝐽))
16		inidm 4149	. . . . . . 7 ⊢ (∪ 𝐽 ∩ ∪ 𝐽) = ∪ 𝐽
17	15, 16	eqtrdi 2795	. . . . . 6 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = ∪ 𝐽)
18	5, 17	sseqtrid 3969	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ⊆ ∪ 𝐽)
19		hauseqcn.e	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
20		ffn 6584	. . . . . . . 8 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → 𝐹 Fn ∪ 𝐽)
21	2, 8, 20	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ∪ 𝐽)
22		ffn 6584	. . . . . . . 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 3967	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
26		fnreseql 6907	. . . . . . 7 ⊢ ((𝐹 Fn ∪ 𝐽 ∧ 𝐺 Fn ∪ 𝐽 ∧ 𝐴 ⊆ ∪ 𝐽) → ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ↔ 𝐴 ⊆ dom (𝐹 ∩ 𝐺)))
27	21, 23, 25, 26	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ↔ 𝐴 ⊆ dom (𝐹 ∩ 𝐺)))
28	19, 27	mpbid 231	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ dom (𝐹 ∩ 𝐺))
29	6	clsss 22113	. . . . 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 1369	. . . 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 22705	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ∈ (Clsd‘𝐽))
34		cldcls 22101	. . . . 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 3963	. . 3 ⊢ (𝜑 → 𝑋 ⊆ dom (𝐹 ∩ 𝐺))
37	1, 36	eqsstrrid 3966	. 2 ⊢ (𝜑 → ∪ 𝐽 ⊆ dom (𝐹 ∩ 𝐺))
38		fneqeql2 6906	. . 3 ⊢ ((𝐹 Fn ∪ 𝐽 ∧ 𝐺 Fn ∪ 𝐽) → (𝐹 = 𝐺 ↔ ∪ 𝐽 ⊆ dom (𝐹 ∩ 𝐺)))
39	21, 23, 38	syl2anc 583	. 2 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ ∪ 𝐽 ⊆ dom (𝐹 ∩ 𝐺)))
40	37, 39	mpbird 256	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108   ∩ cin 3882   ⊆ wss 3883  ∪ cuni 4836  dom cdm 5580   ↾ cres 5582   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  Topctop 21950  Clsdccld 22075  clsccl 22077   Cn ccn 22283  Hauscha 22367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575  df-topgen 17071  df-top 21951  df-topon 21968  df-bases 22004  df-cld 22078  df-cls 22080  df-cn 22286  df-haus 22374  df-tx 22621

This theorem is referenced by:  rrhre  31871