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Theorem hauseqcn 31215
 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 21843 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
42, 3syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
5 dmin 5757 . . . . . 6 dom (𝐹𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺)
6 eqid 2822 . . . . . . . . . 10 𝐽 = 𝐽
7 eqid 2822 . . . . . . . . . 10 𝐾 = 𝐾
86, 7cnf 21849 . . . . . . . . 9 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
9 fdm 6502 . . . . . . . . 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 21849 . . . . . . . . 9 (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺: 𝐽 𝐾)
13 fdm 6502 . . . . . . . . 9 (𝐺: 𝐽 𝐾 → dom 𝐺 = 𝐽)
1411, 12, 133syl 18 . . . . . . . 8 (𝜑 → dom 𝐺 = 𝐽)
1510, 14ineq12d 4164 . . . . . . 7 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = ( 𝐽 𝐽))
16 inidm 4169 . . . . . . 7 ( 𝐽 𝐽) = 𝐽
1715, 16syl6eq 2873 . . . . . 6 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝐽)
185, 17sseqtrid 3994 . . . . 5 (𝜑 → dom (𝐹𝐺) ⊆ 𝐽)
19 hauseqcn.e . . . . . 6 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
20 ffn 6494 . . . . . . . 8 (𝐹: 𝐽 𝐾𝐹 Fn 𝐽)
212, 8, 203syl 18 . . . . . . 7 (𝜑𝐹 Fn 𝐽)
22 ffn 6494 . . . . . . . 8 (𝐺: 𝐽 𝐾𝐺 Fn 𝐽)
2311, 12, 223syl 18 . . . . . . 7 (𝜑𝐺 Fn 𝐽)
24 hauseqcn.a . . . . . . . 8 (𝜑𝐴𝑋)
2524, 1sseqtrdi 3992 . . . . . . 7 (𝜑𝐴 𝐽)
26 fnreseql 6800 . . . . . . 7 ((𝐹 Fn 𝐽𝐺 Fn 𝐽𝐴 𝐽) → ((𝐹𝐴) = (𝐺𝐴) ↔ 𝐴 ⊆ dom (𝐹𝐺)))
2721, 23, 25, 26syl3anc 1368 . . . . . 6 (𝜑 → ((𝐹𝐴) = (𝐺𝐴) ↔ 𝐴 ⊆ dom (𝐹𝐺)))
2819, 27mpbid 235 . . . . 5 (𝜑𝐴 ⊆ dom (𝐹𝐺))
296clsss 21657 . . . . 5 ((𝐽 ∈ Top ∧ dom (𝐹𝐺) ⊆ 𝐽𝐴 ⊆ dom (𝐹𝐺)) → ((cls‘𝐽)‘𝐴) ⊆ ((cls‘𝐽)‘dom (𝐹𝐺)))
304, 18, 28, 29syl3anc 1368 . . . 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 22249 . . . . 5 (𝜑 → dom (𝐹𝐺) ∈ (Clsd‘𝐽))
34 cldcls 21645 . . . . 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 3988 . . 3 (𝜑𝑋 ⊆ dom (𝐹𝐺))
371, 36eqsstrrid 3991 . 2 (𝜑 𝐽 ⊆ dom (𝐹𝐺))
38 fneqeql2 6799 . . 3 ((𝐹 Fn 𝐽𝐺 Fn 𝐽) → (𝐹 = 𝐺 𝐽 ⊆ dom (𝐹𝐺)))
3921, 23, 38syl2anc 587 . 2 (𝜑 → (𝐹 = 𝐺 𝐽 ⊆ dom (𝐹𝐺)))
4037, 39mpbird 260 1 (𝜑𝐹 = 𝐺)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2114   ∩ cin 3907   ⊆ wss 3908  ∪ cuni 4813  dom cdm 5532   ↾ cres 5534   Fn wfn 6329  ⟶wf 6330  ‘cfv 6334  (class class class)co 7140  Topctop 21496  Clsdccld 21619  clsccl 21621   Cn ccn 21827  Hauscha 21911 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-map 8395  df-topgen 16708  df-top 21497  df-topon 21514  df-bases 21549  df-cld 21622  df-cls 21624  df-cn 21830  df-haus 21918  df-tx 22165 This theorem is referenced by:  rrhre  31336
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