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Theorem hauseqcn 29066
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 20755 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
42, 3syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
5 dmin 5145 . . . . . 6 dom (𝐹𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺)
6 eqid 2514 . . . . . . . . . 10 𝐽 = 𝐽
7 eqid 2514 . . . . . . . . . 10 𝐾 = 𝐾
86, 7cnf 20761 . . . . . . . . 9 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
9 fdm 5849 . . . . . . . . 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 20761 . . . . . . . . 9 (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺: 𝐽 𝐾)
13 fdm 5849 . . . . . . . . 9 (𝐺: 𝐽 𝐾 → dom 𝐺 = 𝐽)
1411, 12, 133syl 18 . . . . . . . 8 (𝜑 → dom 𝐺 = 𝐽)
1510, 14ineq12d 3680 . . . . . . 7 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = ( 𝐽 𝐽))
16 inidm 3687 . . . . . . 7 ( 𝐽 𝐽) = 𝐽
1715, 16syl6eq 2564 . . . . . 6 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝐽)
185, 17syl5sseq 3520 . . . . 5 (𝜑 → dom (𝐹𝐺) ⊆ 𝐽)
19 hauseqcn.e . . . . . 6 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
20 ffn 5843 . . . . . . . 8 (𝐹: 𝐽 𝐾𝐹 Fn 𝐽)
212, 8, 203syl 18 . . . . . . 7 (𝜑𝐹 Fn 𝐽)
22 ffn 5843 . . . . . . . 8 (𝐺: 𝐽 𝐾𝐺 Fn 𝐽)
2311, 12, 223syl 18 . . . . . . 7 (𝜑𝐺 Fn 𝐽)
24 hauseqcn.a . . . . . . . 8 (𝜑𝐴𝑋)
2524, 1syl6sseq 3518 . . . . . . 7 (𝜑𝐴 𝐽)
26 fnreseql 6118 . . . . . . 7 ((𝐹 Fn 𝐽𝐺 Fn 𝐽𝐴 𝐽) → ((𝐹𝐴) = (𝐺𝐴) ↔ 𝐴 ⊆ dom (𝐹𝐺)))
2721, 23, 25, 26syl3anc 1317 . . . . . 6 (𝜑 → ((𝐹𝐴) = (𝐺𝐴) ↔ 𝐴 ⊆ dom (𝐹𝐺)))
2819, 27mpbid 220 . . . . 5 (𝜑𝐴 ⊆ dom (𝐹𝐺))
296clsss 20569 . . . . 5 ((𝐽 ∈ Top ∧ dom (𝐹𝐺) ⊆ 𝐽𝐴 ⊆ dom (𝐹𝐺)) → ((cls‘𝐽)‘𝐴) ⊆ ((cls‘𝐽)‘dom (𝐹𝐺)))
304, 18, 28, 29syl3anc 1317 . . . 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 21160 . . . . 5 (𝜑 → dom (𝐹𝐺) ∈ (Clsd‘𝐽))
34 cldcls 20557 . . . . 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 3514 . . 3 (𝜑𝑋 ⊆ dom (𝐹𝐺))
371, 36syl5eqssr 3517 . 2 (𝜑 𝐽 ⊆ dom (𝐹𝐺))
38 fneqeql2 6117 . . 3 ((𝐹 Fn 𝐽𝐺 Fn 𝐽) → (𝐹 = 𝐺 𝐽 ⊆ dom (𝐹𝐺)))
3921, 23, 38syl2anc 690 . 2 (𝜑 → (𝐹 = 𝐺 𝐽 ⊆ dom (𝐹𝐺)))
4037, 39mpbird 245 1 (𝜑𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wcel 1938  cin 3443  wss 3444   cuni 4270  dom cdm 4932  cres 4934   Fn wfn 5684  wf 5685  cfv 5689  (class class class)co 6425  Topctop 20418  Clsdccld 20531  clsccl 20533   Cn ccn 20739  Hauscha 20823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6722
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-1st 6933  df-2nd 6934  df-map 7621  df-topgen 15809  df-top 20422  df-bases 20423  df-topon 20424  df-cld 20534  df-cls 20536  df-cn 20742  df-haus 20830  df-tx 21076
This theorem is referenced by:  rrhre  29190
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