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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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