Theorem hauseqcn 34196

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 23301	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
4	2, 3	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
5		dmin 5888	. . . . . 6 ⊢ dom (𝐹 ∩ 𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺)
6		eqid 2763	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
7		eqid 2763	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
8	6, 7	cnf 23307	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
9		fdm 6702	. . . . . . . . 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 23307	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺:∪ 𝐽⟶∪ 𝐾)
13		fdm 6702	. . . . . . . . 9 ⊢ (𝐺:∪ 𝐽⟶∪ 𝐾 → dom 𝐺 = ∪ 𝐽)
14	11, 12, 13	3syl 18	. . . . . . . 8 ⊢ (𝜑 → dom 𝐺 = ∪ 𝐽)
15	10, 14	ineq12d 4174	. . . . . . 7 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (∪ 𝐽 ∩ ∪ 𝐽))
16		inidm 4179	. . . . . . 7 ⊢ (∪ 𝐽 ∩ ∪ 𝐽) = ∪ 𝐽
17	15, 16	eqtrdi 2814	. . . . . 6 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = ∪ 𝐽)
18	5, 17	sseqtrid 3979	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ⊆ ∪ 𝐽)
19		hauseqcn.e	. . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
20		ffn 6692	. . . . . . . 8 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → 𝐹 Fn ∪ 𝐽)
21	2, 8, 20	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn ∪ 𝐽)
22		ffn 6692	. . . . . . . 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 3977	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽)
26		fnreseql 7030	. . . . . . 7 ⊢ ((𝐹 Fn ∪ 𝐽 ∧ 𝐺 Fn ∪ 𝐽 ∧ 𝐴 ⊆ ∪ 𝐽) → ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ↔ 𝐴 ⊆ dom (𝐹 ∩ 𝐺)))
27	21, 23, 25, 26	syl3anc 1391	. . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ↔ 𝐴 ⊆ dom (𝐹 ∩ 𝐺)))
28	19, 27	mpbid 234	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ dom (𝐹 ∩ 𝐺))
29	6	clsss 23115	. . . . 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 1391	. . . 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 23707	. . . . 5 ⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ∈ (Clsd‘𝐽))
34		cldcls 23103	. . . . 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 3991	. . 3 ⊢ (𝜑 → 𝑋 ⊆ dom (𝐹 ∩ 𝐺))
37	1, 36	eqsstrrid 3976	. 2 ⊢ (𝜑 → ∪ 𝐽 ⊆ dom (𝐹 ∩ 𝐺))
38		fneqeql2 7029	. . 3 ⊢ ((𝐹 Fn ∪ 𝐽 ∧ 𝐺 Fn ∪ 𝐽) → (𝐹 = 𝐺 ↔ ∪ 𝐽 ⊆ dom (𝐹 ∩ 𝐺)))
39	21, 23, 38	syl2anc 593	. 2 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ ∪ 𝐽 ⊆ dom (𝐹 ∩ 𝐺)))
40	37, 39	mpbird 259	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1561   ∈ wcel 2143   ∩ cin 3904   ⊆ wss 3905  ∪ cuni 4866  dom cdm 5648   ↾ cres 5650   Fn wfn 6517  ⟶wf 6518  ‘cfv 6522  (class class class)co 7397  Topctop 22954  Clsdccld 23077  clsccl 23079   Cn ccn 23285  Hauscha 23369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-1st 7971  df-2nd 7972  df-map 8811  df-topgen 17473  df-top 22955  df-topon 22972  df-bases 23007  df-cld 23080  df-cls 23082  df-cn 23288  df-haus 23376  df-tx 23623

This theorem is referenced by:  rrhre  34319