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Theorem hauseqlcld 21362
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 2621 . . . . . . . . . . 11 𝐽 = 𝐽
3 eqid 2621 . . . . . . . . . . 11 𝐾 = 𝐾
42, 3cnf 20963 . . . . . . . . . 10 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
51, 4syl 17 . . . . . . . . 9 (𝜑𝐹: 𝐽 𝐾)
65ffvelrnda 6317 . . . . . . . 8 ((𝜑𝑏 𝐽) → (𝐹𝑏) ∈ 𝐾)
76biantrud 528 . . . . . . 7 ((𝜑𝑏 𝐽) → (⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I ↔ (⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I ∧ (𝐹𝑏) ∈ 𝐾)))
8 fvex 6160 . . . . . . . . 9 (𝐺𝑏) ∈ V
98ideq 5236 . . . . . . . 8 ((𝐹𝑏) I (𝐺𝑏) ↔ (𝐹𝑏) = (𝐺𝑏))
10 df-br 4616 . . . . . . . 8 ((𝐹𝑏) I (𝐺𝑏) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I )
119, 10bitr3i 266 . . . . . . 7 ((𝐹𝑏) = (𝐺𝑏) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I )
128opelres 5363 . . . . . . 7 (⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ ( I ↾ 𝐾) ↔ (⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ I ∧ (𝐹𝑏) ∈ 𝐾))
137, 11, 123bitr4g 303 . . . . . 6 ((𝜑𝑏 𝐽) → ((𝐹𝑏) = (𝐺𝑏) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ ( I ↾ 𝐾)))
14 fveq2 6150 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝐹𝑎) = (𝐹𝑏))
15 fveq2 6150 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝐺𝑎) = (𝐺𝑏))
1614, 15opeq12d 4380 . . . . . . . . 9 (𝑎 = 𝑏 → ⟨(𝐹𝑎), (𝐺𝑎)⟩ = ⟨(𝐹𝑏), (𝐺𝑏)⟩)
17 eqid 2621 . . . . . . . . 9 (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) = (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)
18 opex 4895 . . . . . . . . 9 ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ V
1916, 17, 18fvmpt 6241 . . . . . . . 8 (𝑏 𝐽 → ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) = ⟨(𝐹𝑏), (𝐺𝑏)⟩)
2019adantl 482 . . . . . . 7 ((𝜑𝑏 𝐽) → ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) = ⟨(𝐹𝑏), (𝐺𝑏)⟩)
2120eleq1d 2683 . . . . . 6 ((𝜑𝑏 𝐽) → (((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) ∈ ( I ↾ 𝐾) ↔ ⟨(𝐹𝑏), (𝐺𝑏)⟩ ∈ ( I ↾ 𝐾)))
2213, 21bitr4d 271 . . . . 5 ((𝜑𝑏 𝐽) → ((𝐹𝑏) = (𝐺𝑏) ↔ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) ∈ ( I ↾ 𝐾)))
2322pm5.32da 672 . . . 4 (𝜑 → ((𝑏 𝐽 ∧ (𝐹𝑏) = (𝐺𝑏)) ↔ (𝑏 𝐽 ∧ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) ∈ ( I ↾ 𝐾))))
24 ffn 6004 . . . . . . . 8 (𝐹: 𝐽 𝐾𝐹 Fn 𝐽)
255, 24syl 17 . . . . . . 7 (𝜑𝐹 Fn 𝐽)
26 hauseqlcld.g . . . . . . . . 9 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
272, 3cnf 20963 . . . . . . . . 9 (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺: 𝐽 𝐾)
2826, 27syl 17 . . . . . . . 8 (𝜑𝐺: 𝐽 𝐾)
29 ffn 6004 . . . . . . . 8 (𝐺: 𝐽 𝐾𝐺 Fn 𝐽)
3028, 29syl 17 . . . . . . 7 (𝜑𝐺 Fn 𝐽)
31 fndmin 6282 . . . . . . 7 ((𝐹 Fn 𝐽𝐺 Fn 𝐽) → dom (𝐹𝐺) = {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)})
3225, 30, 31syl2anc 692 . . . . . 6 (𝜑 → dom (𝐹𝐺) = {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)})
3332eleq2d 2684 . . . . 5 (𝜑 → (𝑏 ∈ dom (𝐹𝐺) ↔ 𝑏 ∈ {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)}))
34 rabid 3106 . . . . 5 (𝑏 ∈ {𝑏 𝐽 ∣ (𝐹𝑏) = (𝐺𝑏)} ↔ (𝑏 𝐽 ∧ (𝐹𝑏) = (𝐺𝑏)))
3533, 34syl6bb 276 . . . 4 (𝜑 → (𝑏 ∈ dom (𝐹𝐺) ↔ (𝑏 𝐽 ∧ (𝐹𝑏) = (𝐺𝑏))))
36 opex 4895 . . . . . 6 ⟨(𝐹𝑎), (𝐺𝑎)⟩ ∈ V
3736, 17fnmpti 5981 . . . . 5 (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) Fn 𝐽
38 elpreima 6295 . . . . 5 ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) Fn 𝐽 → (𝑏 ∈ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾)) ↔ (𝑏 𝐽 ∧ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) ∈ ( I ↾ 𝐾))))
3937, 38mp1i 13 . . . 4 (𝜑 → (𝑏 ∈ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾)) ↔ (𝑏 𝐽 ∧ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩)‘𝑏) ∈ ( I ↾ 𝐾))))
4023, 35, 393bitr4d 300 . . 3 (𝜑 → (𝑏 ∈ dom (𝐹𝐺) ↔ 𝑏 ∈ ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾))))
4140eqrdv 2619 . 2 (𝜑 → dom (𝐹𝐺) = ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾)))
422, 17txcnmpt 21340 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐽 Cn 𝐾)) → (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾)))
431, 26, 42syl2anc 692 . . 3 (𝜑 → (𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾)))
44 hauseqlcld.k . . . 4 (𝜑𝐾 ∈ Haus)
453hausdiag 21361 . . . . 5 (𝐾 ∈ Haus ↔ (𝐾 ∈ Top ∧ ( I ↾ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾))))
4645simprbi 480 . . . 4 (𝐾 ∈ Haus → ( I ↾ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))
4744, 46syl 17 . . 3 (𝜑 → ( I ↾ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))
48 cnclima 20985 . . 3 (((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾)) ∧ ( I ↾ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾))) → ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾)) ∈ (Clsd‘𝐽))
4943, 47, 48syl2anc 692 . 2 (𝜑 → ((𝑎 𝐽 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩) “ ( I ↾ 𝐾)) ∈ (Clsd‘𝐽))
5041, 49eqeltrd 2698 1 (𝜑 → dom (𝐹𝐺) ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {crab 2911  cin 3555  cop 4156   cuni 4404   class class class wbr 4615  cmpt 4675   I cid 4986  ccnv 5075  dom cdm 5076  cres 5078  cima 5079   Fn wfn 5844  wf 5845  cfv 5849  (class class class)co 6607  Topctop 20620  Clsdccld 20733   Cn ccn 20941  Hauscha 21025   ×t ctx 21276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-map 7807  df-topgen 16028  df-top 20621  df-topon 20638  df-bases 20664  df-cld 20736  df-cn 20944  df-haus 21032  df-tx 21278
This theorem is referenced by:  hauseqcn  29735  hausgraph  37292
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