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Theorem hauseqlcld 23670
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 2735 . . . . . . . . . . 11 𝐽 = 𝐽
3 eqid 2735 . . . . . . . . . . 11 𝐾 = 𝐾
42, 3cnf 23270 . . . . . . . . . 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 6920 . . . . . . . . 9 (𝐺𝑏) ∈ V
98ideq 5866 . . . . . . . 8 ((𝐹𝑏) I (𝐺𝑏) ↔ (𝐹𝑏) = (𝐺𝑏))
10 df-br 5149 . . . . . . . 8 ((𝐹𝑏) I (𝐺𝑏) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I )
119, 10bitr3i 277 . . . . . . 7 ((𝐹𝑏) = (𝐺𝑏) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I )
128opelresi 6008 . . . . . . 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 6907 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
15 fveq2 6907 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝐺𝑎) = (𝐺𝑏))
1614, 15opeq12d 4886 . . . . . . . . 9 (𝑎 = 𝑏 → ⟨(𝐹𝑎), (𝐺𝑎)⟩ = ⟨(𝐹𝑏), (𝐺𝑏)⟩)
17 eqid 2735 . . . . . . . . 9 (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) = (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)
18 opex 5475 . . . . . . . . 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 2824 . . . . . 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 6738 . . . . . . 7 (𝜑𝐹 Fn 𝐽)
25 hauseqlcld.g . . . . . . . . 9 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
262, 3cnf 23270 . . . . . . . . 9 (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺: 𝐽 𝐾)
2725, 26syl 17 . . . . . . . 8 (𝜑𝐺: 𝐽 𝐾)
2827ffnd 6738 . . . . . . 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 2825 . . . . 5 (𝜑 → (𝑏 ∈ dom (𝐹𝐺) ↔ 𝑏 ∈ {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)}))
32 rabid 3455 . . . . 5 (𝑏 ∈ {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)} ↔ (𝑏 𝐽 ∧ (𝐹𝑏) = (𝐺𝑏)))
3331, 32bitrdi 287 . . . 4 (𝜑 → (𝑏 ∈ dom (𝐹𝐺) ↔ (𝑏 𝐽 ∧ (𝐹𝑏) = (𝐺𝑏))))
34 opex 5475 . . . . . 6 ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ V
3534, 17fnmpti 6712 . . . . 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 2733 . 2 (𝜑 → dom (𝐹𝐺) = ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾)))
402, 17txcnmpt 23648 . . . 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 23669 . . . . 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 23292 . . 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 2839 1 (𝜑 → dom (𝐹𝐺) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {crab 3433  cin 3962  cop 4637   cuni 4912   class class class wbr 5148  cmpt 5231   I cid 5582  ccnv 5688  dom cdm 5689  cres 5691  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  Topctop 22915  Clsdccld 23040   Cn ccn 23248  Hauscha 23332   ×t ctx 23584
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-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-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-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-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-cn 23251  df-haus 23339  df-tx 23586
This theorem is referenced by:  hauseqcn  33859  hausgraph  43194
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