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Theorem hauseqlcld 23149
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 2732 . . . . . . . . . . 11 𝐽 = 𝐽
3 eqid 2732 . . . . . . . . . . 11 𝐾 = 𝐾
42, 3cnf 22749 . . . . . . . . . 10 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
51, 4syl 17 . . . . . . . . 9 (𝜑𝐹: 𝐽 𝐾)
65ffvelcdmda 7086 . . . . . . . 8 ((𝜑𝑏 𝐽) → (𝐹𝑏) ∈ 𝐾)
76biantrurd 533 . . . . . . 7 ((𝜑𝑏 𝐽) → (⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I ↔ ((𝐹𝑏) ∈ 𝐾 ∧ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I )))
8 fvex 6904 . . . . . . . . 9 (𝐺𝑏) ∈ V
98ideq 5852 . . . . . . . 8 ((𝐹𝑏) I (𝐺𝑏) ↔ (𝐹𝑏) = (𝐺𝑏))
10 df-br 5149 . . . . . . . 8 ((𝐹𝑏) I (𝐺𝑏) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I )
119, 10bitr3i 276 . . . . . . 7 ((𝐹𝑏) = (𝐺𝑏) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I )
128opelresi 5989 . . . . . . 7 (⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ ( I ↾ 𝐾) ↔ ((𝐹𝑏) ∈ 𝐾 ∧ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I ))
137, 11, 123bitr4g 313 . . . . . 6 ((𝜑𝑏 𝐽) → ((𝐹𝑏) = (𝐺𝑏) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ ( I ↾ 𝐾)))
14 fveq2 6891 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
15 fveq2 6891 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝐺𝑎) = (𝐺𝑏))
1614, 15opeq12d 4881 . . . . . . . . 9 (𝑎 = 𝑏 → ⟨(𝐹𝑎), (𝐺𝑎)⟩ = ⟨(𝐹𝑏), (𝐺𝑏)⟩)
17 eqid 2732 . . . . . . . . 9 (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) = (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)
18 opex 5464 . . . . . . . . 9 ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ V
1916, 17, 18fvmpt 6998 . . . . . . . 8 (𝑏 𝐽 → ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) = ⟨(𝐹𝑏), (𝐺𝑏)⟩)
2019adantl 482 . . . . . . 7 ((𝜑𝑏 𝐽) → ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) = ⟨(𝐹𝑏), (𝐺𝑏)⟩)
2120eleq1d 2818 . . . . . 6 ((𝜑𝑏 𝐽) → (((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) ∈ ( I ↾ 𝐾) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ ( I ↾ 𝐾)))
2213, 21bitr4d 281 . . . . 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 6718 . . . . . . 7 (𝜑𝐹 Fn 𝐽)
25 hauseqlcld.g . . . . . . . . 9 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
262, 3cnf 22749 . . . . . . . . 9 (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺: 𝐽 𝐾)
2725, 26syl 17 . . . . . . . 8 (𝜑𝐺: 𝐽 𝐾)
2827ffnd 6718 . . . . . . 7 (𝜑𝐺 Fn 𝐽)
29 fndmin 7046 . . . . . . 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 2819 . . . . 5 (𝜑 → (𝑏 ∈ dom (𝐹𝐺) ↔ 𝑏 ∈ {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)}))
32 rabid 3452 . . . . 5 (𝑏 ∈ {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)} ↔ (𝑏 𝐽 ∧ (𝐹𝑏) = (𝐺𝑏)))
3331, 32bitrdi 286 . . . 4 (𝜑 → (𝑏 ∈ dom (𝐹𝐺) ↔ (𝑏 𝐽 ∧ (𝐹𝑏) = (𝐺𝑏))))
34 opex 5464 . . . . . 6 ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ V
3534, 17fnmpti 6693 . . . . 5 (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) Fn 𝐽
36 elpreima 7059 . . . . 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 310 . . 3 (𝜑 → (𝑏 ∈ dom (𝐹𝐺) ↔ 𝑏 ∈ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾))))
3938eqrdv 2730 . 2 (𝜑 → dom (𝐹𝐺) = ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾)))
402, 17txcnmpt 23127 . . . 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 23148 . . . . 5 (𝐾 ∈ Haus ↔ (𝐾 ∈ Top ∧ ( I ↾ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾))))
4443simprbi 497 . . . 4 (𝐾 ∈ Haus → ( I ↾ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))
4542, 44syl 17 . . 3 (𝜑 → ( I ↾ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))
46 cnclima 22771 . . 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 2833 1 (𝜑 → dom (𝐹𝐺) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {crab 3432  cin 3947  cop 4634   cuni 4908   class class class wbr 5148  cmpt 5231   I cid 5573  ccnv 5675  dom cdm 5676  cres 5678  cima 5679   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7408  Topctop 22394  Clsdccld 22519   Cn ccn 22727  Hauscha 22811   ×t ctx 23063
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-map 8821  df-topgen 17388  df-top 22395  df-topon 22412  df-bases 22448  df-cld 22522  df-cn 22730  df-haus 22818  df-tx 23065
This theorem is referenced by:  hauseqcn  32873  hausgraph  41944
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