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Theorem hauseqcn 33859
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 23264 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
42, 3syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
5 dmin 5925 . . . . . 6 dom (𝐹𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺)
6 eqid 2735 . . . . . . . . . 10 𝐽 = 𝐽
7 eqid 2735 . . . . . . . . . 10 𝐾 = 𝐾
86, 7cnf 23270 . . . . . . . . 9 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
9 fdm 6746 . . . . . . . . 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 23270 . . . . . . . . 9 (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺: 𝐽 𝐾)
13 fdm 6746 . . . . . . . . 9 (𝐺: 𝐽 𝐾 → dom 𝐺 = 𝐽)
1411, 12, 133syl 18 . . . . . . . 8 (𝜑 → dom 𝐺 = 𝐽)
1510, 14ineq12d 4229 . . . . . . 7 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = ( 𝐽 𝐽))
16 inidm 4235 . . . . . . 7 ( 𝐽 𝐽) = 𝐽
1715, 16eqtrdi 2791 . . . . . 6 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝐽)
185, 17sseqtrid 4048 . . . . 5 (𝜑 → dom (𝐹𝐺) ⊆ 𝐽)
19 hauseqcn.e . . . . . 6 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
20 ffn 6737 . . . . . . . 8 (𝐹: 𝐽 𝐾𝐹 Fn 𝐽)
212, 8, 203syl 18 . . . . . . 7 (𝜑𝐹 Fn 𝐽)
22 ffn 6737 . . . . . . . 8 (𝐺: 𝐽 𝐾𝐺 Fn 𝐽)
2311, 12, 223syl 18 . . . . . . 7 (𝜑𝐺 Fn 𝐽)
24 hauseqcn.a . . . . . . . 8 (𝜑𝐴𝑋)
2524, 1sseqtrdi 4046 . . . . . . 7 (𝜑𝐴 𝐽)
26 fnreseql 7068 . . . . . . 7 ((𝐹 Fn 𝐽𝐺 Fn 𝐽𝐴 𝐽) → ((𝐹𝐴) = (𝐺𝐴) ↔ 𝐴 ⊆ dom (𝐹𝐺)))
2721, 23, 25, 26syl3anc 1370 . . . . . 6 (𝜑 → ((𝐹𝐴) = (𝐺𝐴) ↔ 𝐴 ⊆ dom (𝐹𝐺)))
2819, 27mpbid 232 . . . . 5 (𝜑𝐴 ⊆ dom (𝐹𝐺))
296clsss 23078 . . . . 5 ((𝐽 ∈ Top ∧ dom (𝐹𝐺) ⊆ 𝐽𝐴 ⊆ dom (𝐹𝐺)) → ((cls‘𝐽)‘𝐴) ⊆ ((cls‘𝐽)‘dom (𝐹𝐺)))
304, 18, 28, 29syl3anc 1370 . . . 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 23670 . . . . 5 (𝜑 → dom (𝐹𝐺) ∈ (Clsd‘𝐽))
34 cldcls 23066 . . . . 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 4042 . . 3 (𝜑𝑋 ⊆ dom (𝐹𝐺))
371, 36eqsstrrid 4045 . 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 1537  wcel 2106  cin 3962  wss 3963   cuni 4912  dom cdm 5689  cres 5691   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  Topctop 22915  Clsdccld 23040  clsccl 23042   Cn ccn 23248  Hauscha 23332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-cls 23045  df-cn 23251  df-haus 23339  df-tx 23586
This theorem is referenced by:  rrhre  33984
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