MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hauseqlcld Structured version   Visualization version   GIF version

Theorem hauseqlcld 23150
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 2733 . . . . . . . . . . 11 𝐽 = 𝐽
3 eqid 2733 . . . . . . . . . . 11 𝐾 = 𝐾
42, 3cnf 22750 . . . . . . . . . 10 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
51, 4syl 17 . . . . . . . . 9 (𝜑𝐹: 𝐽 𝐾)
65ffvelcdmda 7087 . . . . . . . 8 ((𝜑𝑏 𝐽) → (𝐹𝑏) ∈ 𝐾)
76biantrurd 534 . . . . . . 7 ((𝜑𝑏 𝐽) → (⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I ↔ ((𝐹𝑏) ∈ 𝐾 ∧ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I )))
8 fvex 6905 . . . . . . . . 9 (𝐺𝑏) ∈ V
98ideq 5853 . . . . . . . 8 ((𝐹𝑏) I (𝐺𝑏) ↔ (𝐹𝑏) = (𝐺𝑏))
10 df-br 5150 . . . . . . . 8 ((𝐹𝑏) I (𝐺𝑏) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I )
119, 10bitr3i 277 . . . . . . 7 ((𝐹𝑏) = (𝐺𝑏) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I )
128opelresi 5990 . . . . . . 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 6892 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
15 fveq2 6892 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝐺𝑎) = (𝐺𝑏))
1614, 15opeq12d 4882 . . . . . . . . 9 (𝑎 = 𝑏 → ⟨(𝐹𝑎), (𝐺𝑎)⟩ = ⟨(𝐹𝑏), (𝐺𝑏)⟩)
17 eqid 2733 . . . . . . . . 9 (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) = (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)
18 opex 5465 . . . . . . . . 9 ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ V
1916, 17, 18fvmpt 6999 . . . . . . . 8 (𝑏 𝐽 → ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) = ⟨(𝐹𝑏), (𝐺𝑏)⟩)
2019adantl 483 . . . . . . 7 ((𝜑𝑏 𝐽) → ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) = ⟨(𝐹𝑏), (𝐺𝑏)⟩)
2120eleq1d 2819 . . . . . 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 580 . . . 4 (𝜑 → ((𝑏 𝐽 ∧ (𝐹𝑏) = (𝐺𝑏)) ↔ (𝑏 𝐽 ∧ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) ∈ ( I ↾ 𝐾))))
245ffnd 6719 . . . . . . 7 (𝜑𝐹 Fn 𝐽)
25 hauseqlcld.g . . . . . . . . 9 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
262, 3cnf 22750 . . . . . . . . 9 (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺: 𝐽 𝐾)
2725, 26syl 17 . . . . . . . 8 (𝜑𝐺: 𝐽 𝐾)
2827ffnd 6719 . . . . . . 7 (𝜑𝐺 Fn 𝐽)
29 fndmin 7047 . . . . . . 7 ((𝐹 Fn 𝐽𝐺 Fn 𝐽) → dom (𝐹𝐺) = {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)})
3024, 28, 29syl2anc 585 . . . . . 6 (𝜑 → dom (𝐹𝐺) = {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)})
3130eleq2d 2820 . . . . 5 (𝜑 → (𝑏 ∈ dom (𝐹𝐺) ↔ 𝑏 ∈ {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)}))
32 rabid 3453 . . . . 5 (𝑏 ∈ {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)} ↔ (𝑏 𝐽 ∧ (𝐹𝑏) = (𝐺𝑏)))
3331, 32bitrdi 287 . . . 4 (𝜑 → (𝑏 ∈ dom (𝐹𝐺) ↔ (𝑏 𝐽 ∧ (𝐹𝑏) = (𝐺𝑏))))
34 opex 5465 . . . . . 6 ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ V
3534, 17fnmpti 6694 . . . . 5 (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) Fn 𝐽
36 elpreima 7060 . . . . 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 2731 . 2 (𝜑 → dom (𝐹𝐺) = ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾)))
402, 17txcnmpt 23128 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐽 Cn 𝐾)) → (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾)))
411, 25, 40syl2anc 585 . . 3 (𝜑 → (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾)))
42 hauseqlcld.k . . . 4 (𝜑𝐾 ∈ Haus)
433hausdiag 23149 . . . . 5 (𝐾 ∈ Haus ↔ (𝐾 ∈ Top ∧ ( I ↾ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾))))
4443simprbi 498 . . . 4 (𝐾 ∈ Haus → ( I ↾ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))
4542, 44syl 17 . . 3 (𝜑 → ( I ↾ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))
46 cnclima 22772 . . 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 585 . 2 (𝜑 → ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾)) ∈ (Clsd‘𝐽))
4839, 47eqeltrd 2834 1 (𝜑 → dom (𝐹𝐺) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {crab 3433  cin 3948  cop 4635   cuni 4909   class class class wbr 5149  cmpt 5232   I cid 5574  ccnv 5676  dom cdm 5677  cres 5679  cima 5680   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7409  Topctop 22395  Clsdccld 22520   Cn ccn 22728  Hauscha 22812   ×t ctx 23064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-topgen 17389  df-top 22396  df-topon 22413  df-bases 22449  df-cld 22523  df-cn 22731  df-haus 22819  df-tx 23066
This theorem is referenced by:  hauseqcn  32878  hausgraph  41954
  Copyright terms: Public domain W3C validator