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Theorem hauseqlcld 23654
Description: In a Hausdorff topology, the equalizer of two continuous functions is closed (thus, two continuous functions which agree on a dense set agree everywhere). (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
hauseqlcld.k (𝜑𝐾 ∈ Haus)
hauseqlcld.f (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
hauseqlcld.g (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
Assertion
Ref Expression
hauseqlcld (𝜑 → dom (𝐹𝐺) ∈ (Clsd‘𝐽))

Proof of Theorem hauseqlcld
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hauseqlcld.f . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
2 eqid 2737 . . . . . . . . . . 11 𝐽 = 𝐽
3 eqid 2737 . . . . . . . . . . 11 𝐾 = 𝐾
42, 3cnf 23254 . . . . . . . . . 10 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
51, 4syl 17 . . . . . . . . 9 (𝜑𝐹: 𝐽 𝐾)
65ffvelcdmda 7104 . . . . . . . 8 ((𝜑𝑏 𝐽) → (𝐹𝑏) ∈ 𝐾)
76biantrurd 532 . . . . . . 7 ((𝜑𝑏 𝐽) → (⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I ↔ ((𝐹𝑏) ∈ 𝐾 ∧ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I )))
8 fvex 6919 . . . . . . . . 9 (𝐺𝑏) ∈ V
98ideq 5863 . . . . . . . 8 ((𝐹𝑏) I (𝐺𝑏) ↔ (𝐹𝑏) = (𝐺𝑏))
10 df-br 5144 . . . . . . . 8 ((𝐹𝑏) I (𝐺𝑏) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I )
119, 10bitr3i 277 . . . . . . 7 ((𝐹𝑏) = (𝐺𝑏) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I )
128opelresi 6005 . . . . . . 7 (⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ ( I ↾ 𝐾) ↔ ((𝐹𝑏) ∈ 𝐾 ∧ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I ))
137, 11, 123bitr4g 314 . . . . . 6 ((𝜑𝑏 𝐽) → ((𝐹𝑏) = (𝐺𝑏) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ ( I ↾ 𝐾)))
14 fveq2 6906 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
15 fveq2 6906 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝐺𝑎) = (𝐺𝑏))
1614, 15opeq12d 4881 . . . . . . . . 9 (𝑎 = 𝑏 → ⟨(𝐹𝑎), (𝐺𝑎)⟩ = ⟨(𝐹𝑏), (𝐺𝑏)⟩)
17 eqid 2737 . . . . . . . . 9 (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) = (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)
18 opex 5469 . . . . . . . . 9 ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ V
1916, 17, 18fvmpt 7016 . . . . . . . 8 (𝑏 𝐽 → ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) = ⟨(𝐹𝑏), (𝐺𝑏)⟩)
2019adantl 481 . . . . . . 7 ((𝜑𝑏 𝐽) → ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) = ⟨(𝐹𝑏), (𝐺𝑏)⟩)
2120eleq1d 2826 . . . . . 6 ((𝜑𝑏 𝐽) → (((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) ∈ ( I ↾ 𝐾) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ ( I ↾ 𝐾)))
2213, 21bitr4d 282 . . . . 5 ((𝜑𝑏 𝐽) → ((𝐹𝑏) = (𝐺𝑏) ↔ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) ∈ ( I ↾ 𝐾)))
2322pm5.32da 579 . . . 4 (𝜑 → ((𝑏 𝐽 ∧ (𝐹𝑏) = (𝐺𝑏)) ↔ (𝑏 𝐽 ∧ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) ∈ ( I ↾ 𝐾))))
245ffnd 6737 . . . . . . 7 (𝜑𝐹 Fn 𝐽)
25 hauseqlcld.g . . . . . . . . 9 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
262, 3cnf 23254 . . . . . . . . 9 (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺: 𝐽 𝐾)
2725, 26syl 17 . . . . . . . 8 (𝜑𝐺: 𝐽 𝐾)
2827ffnd 6737 . . . . . . 7 (𝜑𝐺 Fn 𝐽)
29 fndmin 7065 . . . . . . 7 ((𝐹 Fn 𝐽𝐺 Fn 𝐽) → dom (𝐹𝐺) = {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)})
3024, 28, 29syl2anc 584 . . . . . 6 (𝜑 → dom (𝐹𝐺) = {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)})
3130eleq2d 2827 . . . . 5 (𝜑 → (𝑏 ∈ dom (𝐹𝐺) ↔ 𝑏 ∈ {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)}))
32 rabid 3458 . . . . 5 (𝑏 ∈ {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)} ↔ (𝑏 𝐽 ∧ (𝐹𝑏) = (𝐺𝑏)))
3331, 32bitrdi 287 . . . 4 (𝜑 → (𝑏 ∈ dom (𝐹𝐺) ↔ (𝑏 𝐽 ∧ (𝐹𝑏) = (𝐺𝑏))))
34 opex 5469 . . . . . 6 ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ V
3534, 17fnmpti 6711 . . . . 5 (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) Fn 𝐽
36 elpreima 7078 . . . . 5 ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) Fn 𝐽 → (𝑏 ∈ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾)) ↔ (𝑏 𝐽 ∧ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) ∈ ( I ↾ 𝐾))))
3735, 36mp1i 13 . . . 4 (𝜑 → (𝑏 ∈ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾)) ↔ (𝑏 𝐽 ∧ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) ∈ ( I ↾ 𝐾))))
3823, 33, 373bitr4d 311 . . 3 (𝜑 → (𝑏 ∈ dom (𝐹𝐺) ↔ 𝑏 ∈ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾))))
3938eqrdv 2735 . 2 (𝜑 → dom (𝐹𝐺) = ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾)))
402, 17txcnmpt 23632 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐽 Cn 𝐾)) → (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾)))
411, 25, 40syl2anc 584 . . 3 (𝜑 → (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾)))
42 hauseqlcld.k . . . 4 (𝜑𝐾 ∈ Haus)
433hausdiag 23653 . . . . 5 (𝐾 ∈ Haus ↔ (𝐾 ∈ Top ∧ ( I ↾ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾))))
4443simprbi 496 . . . 4 (𝐾 ∈ Haus → ( I ↾ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))
4542, 44syl 17 . . 3 (𝜑 → ( I ↾ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))
46 cnclima 23276 . . 3 (((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾)) ∧ ( I ↾ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾))) → ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾)) ∈ (Clsd‘𝐽))
4741, 45, 46syl2anc 584 . 2 (𝜑 → ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾)) ∈ (Clsd‘𝐽))
4839, 47eqeltrd 2841 1 (𝜑 → dom (𝐹𝐺) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {crab 3436  cin 3950  cop 4632   cuni 4907   class class class wbr 5143  cmpt 5225   I cid 5577  ccnv 5684  dom cdm 5685  cres 5687  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Topctop 22899  Clsdccld 23024   Cn ccn 23232  Hauscha 23316   ×t ctx 23568
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-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-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-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-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-cn 23235  df-haus 23323  df-tx 23570
This theorem is referenced by:  hauseqcn  33897  hausgraph  43217
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