Theorem hauseqcn 34530

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 23558	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
4	2, 3	syl 18	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
5		dmin 5893	. . . . . 6 ⊢ dom (𝐹 ∩ 𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺)
6		eqid 2761	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
7		eqid 2761	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
8	6, 7	cnf 23564	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
9		fdm 6719	. . . . . . . . 9 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → dom 𝐹 = ∪ 𝐽)
10	2, 8, 9	3syl 19	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = ∪ 𝐽)
11		hauseqcn.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
12	6, 7	cnf 23564	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺:∪ 𝐽⟶∪ 𝐾)
13		fdm 6719	. . . . . . . . 9 ⊢ (𝐺:∪ 𝐽⟶∪ 𝐾 → dom 𝐺 = ∪ 𝐽)
14	11, 12, 13	3syl 19	. . . . . . . 8 ⊢ (𝜑 → dom 𝐺 = ∪ 𝐽)
15	10, 14	ineq12d 4167	. . . . . . 7 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (∪ 𝐽 ∩ ∪ 𝐽))
16		inidm 4172	. . . . . . 7 ⊢ (∪ 𝐽 ∩ ∪ 𝐽) = ∪ 𝐽
17	15, 16	eqtrdi 2812	. . . . . 6 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = ∪ 𝐽)
18	5, 17	sseqtrid 3973	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ⊆ ∪ 𝐽)
19		hauseqcn.e	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
20		ffn 6709	. . . . . . . 8 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → 𝐹 Fn ∪ 𝐽)
21	2, 8, 20	3syl 19	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ∪ 𝐽)
22		ffn 6709	. . . . . . . 8 ⊢ (𝐺:∪ 𝐽⟶∪ 𝐾 → 𝐺 Fn ∪ 𝐽)
23	11, 12, 22	3syl 19	. . . . . . 7 ⊢ (𝜑 → 𝐺 Fn ∪ 𝐽)
24		hauseqcn.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
25	24, 1	sseqtrdi 3971	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
26		fnreseql 7047	. . . . . . 7 ⊢ ((𝐹 Fn ∪ 𝐽 ∧ 𝐺 Fn ∪ 𝐽 ∧ 𝐴 ⊆ ∪ 𝐽) → ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ↔ 𝐴 ⊆ dom (𝐹 ∩ 𝐺)))
27	21, 23, 25, 26	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ↔ 𝐴 ⊆ dom (𝐹 ∩ 𝐺)))
28	19, 27	mpbid 235	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ dom (𝐹 ∩ 𝐺))
29	6	clsss 23372	. . . . 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 1398	. . . 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 23965	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ∈ (Clsd‘𝐽))
34		cldcls 23360	. . . . 5 ⊢ (dom (𝐹 ∩ 𝐺) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘dom (𝐹 ∩ 𝐺)) = dom (𝐹 ∩ 𝐺))
35	33, 34	syl 18	. . . 4 ⊢ (𝜑 → ((cls‘𝐽)‘dom (𝐹 ∩ 𝐺)) = dom (𝐹 ∩ 𝐺))
36	30, 31, 35	3sstr3d 3985	. . 3 ⊢ (𝜑 → 𝑋 ⊆ dom (𝐹 ∩ 𝐺))
37	1, 36	eqsstrrid 3970	. 2 ⊢ (𝜑 → ∪ 𝐽 ⊆ dom (𝐹 ∩ 𝐺))
38		fneqeql2 7046	. . 3 ⊢ ((𝐹 Fn ∪ 𝐽 ∧ 𝐺 Fn ∪ 𝐽) → (𝐹 = 𝐺 ↔ ∪ 𝐽 ⊆ dom (𝐹 ∩ 𝐺)))
39	21, 23, 38	syl2anc 596	. 2 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ ∪ 𝐽 ⊆ dom (𝐹 ∩ 𝐺)))
40	37, 39	mpbird 260	1 ⊢ (𝜑 → 𝐹 = 𝐺)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145   ∩ cin 3898   ⊆ wss 3899  ∪ cuni 4867  dom cdm 5651   ↾ cres 5653   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  Topctop 23211  Clsdccld 23334  clsccl 23336   Cn ccn 23542  Hauscha 23626

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8849  df-topgen 17614  df-top 23212  df-topon 23229  df-bases 23264  df-cld 23337  df-cls 23339  df-cn 23545  df-haus 23633  df-tx 23881

This theorem is used by:  rrhre  34653