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Theorem hauseqcn 33897
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 (𝜑𝐾 ∈ Haus)
hauseqcn.f (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
hauseqcn.g (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
hauseqcn.e (𝜑 → (𝐹𝐴) = (𝐺𝐴))
hauseqcn.a (𝜑𝐴𝑋)
hauseqcn.c (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
Assertion
Ref Expression
hauseqcn (𝜑𝐹 = 𝐺)

Proof of Theorem hauseqcn
StepHypRef Expression
1 hauseqcn.x . . 3 𝑋 = 𝐽
2 hauseqcn.f . . . . . 6 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
3 cntop1 23248 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
42, 3syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
5 dmin 5922 . . . . . 6 dom (𝐹𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺)
6 eqid 2737 . . . . . . . . . 10 𝐽 = 𝐽
7 eqid 2737 . . . . . . . . . 10 𝐾 = 𝐾
86, 7cnf 23254 . . . . . . . . 9 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
9 fdm 6745 . . . . . . . . 9 (𝐹: 𝐽 𝐾 → dom 𝐹 = 𝐽)
102, 8, 93syl 18 . . . . . . . 8 (𝜑 → dom 𝐹 = 𝐽)
11 hauseqcn.g . . . . . . . . 9 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
126, 7cnf 23254 . . . . . . . . 9 (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺: 𝐽 𝐾)
13 fdm 6745 . . . . . . . . 9 (𝐺: 𝐽 𝐾 → dom 𝐺 = 𝐽)
1411, 12, 133syl 18 . . . . . . . 8 (𝜑 → dom 𝐺 = 𝐽)
1510, 14ineq12d 4221 . . . . . . 7 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = ( 𝐽 𝐽))
16 inidm 4227 . . . . . . 7 ( 𝐽 𝐽) = 𝐽
1715, 16eqtrdi 2793 . . . . . 6 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝐽)
185, 17sseqtrid 4026 . . . . 5 (𝜑 → dom (𝐹𝐺) ⊆ 𝐽)
19 hauseqcn.e . . . . . 6 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
20 ffn 6736 . . . . . . . 8 (𝐹: 𝐽 𝐾𝐹 Fn 𝐽)
212, 8, 203syl 18 . . . . . . 7 (𝜑𝐹 Fn 𝐽)
22 ffn 6736 . . . . . . . 8 (𝐺: 𝐽 𝐾𝐺 Fn 𝐽)
2311, 12, 223syl 18 . . . . . . 7 (𝜑𝐺 Fn 𝐽)
24 hauseqcn.a . . . . . . . 8 (𝜑𝐴𝑋)
2524, 1sseqtrdi 4024 . . . . . . 7 (𝜑𝐴 𝐽)
26 fnreseql 7068 . . . . . . 7 ((𝐹 Fn 𝐽𝐺 Fn 𝐽𝐴 𝐽) → ((𝐹𝐴) = (𝐺𝐴) ↔ 𝐴 ⊆ dom (𝐹𝐺)))
2721, 23, 25, 26syl3anc 1373 . . . . . 6 (𝜑 → ((𝐹𝐴) = (𝐺𝐴) ↔ 𝐴 ⊆ dom (𝐹𝐺)))
2819, 27mpbid 232 . . . . 5 (𝜑𝐴 ⊆ dom (𝐹𝐺))
296clsss 23062 . . . . 5 ((𝐽 ∈ Top ∧ dom (𝐹𝐺) ⊆ 𝐽𝐴 ⊆ dom (𝐹𝐺)) → ((cls‘𝐽)‘𝐴) ⊆ ((cls‘𝐽)‘dom (𝐹𝐺)))
304, 18, 28, 29syl3anc 1373 . . . 4 (𝜑 → ((cls‘𝐽)‘𝐴) ⊆ ((cls‘𝐽)‘dom (𝐹𝐺)))
31 hauseqcn.c . . . 4 (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)
32 hauseqcn.k . . . . . 6 (𝜑𝐾 ∈ Haus)
3332, 2, 11hauseqlcld 23654 . . . . 5 (𝜑 → dom (𝐹𝐺) ∈ (Clsd‘𝐽))
34 cldcls 23050 . . . . 5 (dom (𝐹𝐺) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘dom (𝐹𝐺)) = dom (𝐹𝐺))
3533, 34syl 17 . . . 4 (𝜑 → ((cls‘𝐽)‘dom (𝐹𝐺)) = dom (𝐹𝐺))
3630, 31, 353sstr3d 4038 . . 3 (𝜑𝑋 ⊆ dom (𝐹𝐺))
371, 36eqsstrrid 4023 . 2 (𝜑 𝐽 ⊆ dom (𝐹𝐺))
38 fneqeql2 7067 . . 3 ((𝐹 Fn 𝐽𝐺 Fn 𝐽) → (𝐹 = 𝐺 𝐽 ⊆ dom (𝐹𝐺)))
3921, 23, 38syl2anc 584 . 2 (𝜑 → (𝐹 = 𝐺 𝐽 ⊆ dom (𝐹𝐺)))
4037, 39mpbird 257 1 (𝜑𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  cin 3950  wss 3951   cuni 4907  dom cdm 5685  cres 5687   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Topctop 22899  Clsdccld 23024  clsccl 23026   Cn ccn 23232  Hauscha 23316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-cls 23029  df-cn 23235  df-haus 23323  df-tx 23570
This theorem is referenced by:  rrhre  34022
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