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Theorem hausgraph 43217
Description: The graph of a continuous function into a Hausdorff space is closed. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Assertion
Ref Expression
hausgraph ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (Clsd‘(𝐽 ×t 𝐾)))

Proof of Theorem hausgraph
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 f1stres 8038 . . . . . . . . 9 (1st ↾ ( 𝐽 × 𝐾)):( 𝐽 × 𝐾)⟶ 𝐽
2 ffn 6736 . . . . . . . . 9 ((1st ↾ ( 𝐽 × 𝐾)):( 𝐽 × 𝐾)⟶ 𝐽 → (1st ↾ ( 𝐽 × 𝐾)) Fn ( 𝐽 × 𝐾))
31, 2ax-mp 5 . . . . . . . 8 (1st ↾ ( 𝐽 × 𝐾)) Fn ( 𝐽 × 𝐾)
4 fvco2 7006 . . . . . . . 8 (((1st ↾ ( 𝐽 × 𝐾)) Fn ( 𝐽 × 𝐾) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = (𝐹‘((1st ↾ ( 𝐽 × 𝐾))‘𝑎)))
53, 4mpan 690 . . . . . . 7 (𝑎 ∈ ( 𝐽 × 𝐾) → ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = (𝐹‘((1st ↾ ( 𝐽 × 𝐾))‘𝑎)))
65adantl 481 . . . . . 6 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = (𝐹‘((1st ↾ ( 𝐽 × 𝐾))‘𝑎)))
7 fvres 6925 . . . . . . . 8 (𝑎 ∈ ( 𝐽 × 𝐾) → ((1st ↾ ( 𝐽 × 𝐾))‘𝑎) = (1st𝑎))
87fveq2d 6910 . . . . . . 7 (𝑎 ∈ ( 𝐽 × 𝐾) → (𝐹‘((1st ↾ ( 𝐽 × 𝐾))‘𝑎)) = (𝐹‘(1st𝑎)))
98adantl 481 . . . . . 6 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → (𝐹‘((1st ↾ ( 𝐽 × 𝐾))‘𝑎)) = (𝐹‘(1st𝑎)))
106, 9eqtrd 2777 . . . . 5 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = (𝐹‘(1st𝑎)))
11 fvres 6925 . . . . . 6 (𝑎 ∈ ( 𝐽 × 𝐾) → ((2nd ↾ ( 𝐽 × 𝐾))‘𝑎) = (2nd𝑎))
1211adantl 481 . . . . 5 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → ((2nd ↾ ( 𝐽 × 𝐾))‘𝑎) = (2nd𝑎))
1310, 12eqeq12d 2753 . . . 4 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → (((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = ((2nd ↾ ( 𝐽 × 𝐾))‘𝑎) ↔ (𝐹‘(1st𝑎)) = (2nd𝑎)))
1413rabbidva 3443 . . 3 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → {𝑎 ∈ ( 𝐽 × 𝐾) ∣ ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = ((2nd ↾ ( 𝐽 × 𝐾))‘𝑎)} = {𝑎 ∈ ( 𝐽 × 𝐾) ∣ (𝐹‘(1st𝑎)) = (2nd𝑎)})
15 eqid 2737 . . . . . . . 8 𝐽 = 𝐽
16 eqid 2737 . . . . . . . 8 𝐾 = 𝐾
1715, 16cnf 23254 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
1817adantl 481 . . . . . 6 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽 𝐾)
19 fco 6760 . . . . . 6 ((𝐹: 𝐽 𝐾 ∧ (1st ↾ ( 𝐽 × 𝐾)):( 𝐽 × 𝐾)⟶ 𝐽) → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))):( 𝐽 × 𝐾)⟶ 𝐾)
2018, 1, 19sylancl 586 . . . . 5 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))):( 𝐽 × 𝐾)⟶ 𝐾)
2120ffnd 6737 . . . 4 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) Fn ( 𝐽 × 𝐾))
22 f2ndres 8039 . . . . 5 (2nd ↾ ( 𝐽 × 𝐾)):( 𝐽 × 𝐾)⟶ 𝐾
23 ffn 6736 . . . . 5 ((2nd ↾ ( 𝐽 × 𝐾)):( 𝐽 × 𝐾)⟶ 𝐾 → (2nd ↾ ( 𝐽 × 𝐾)) Fn ( 𝐽 × 𝐾))
2422, 23ax-mp 5 . . . 4 (2nd ↾ ( 𝐽 × 𝐾)) Fn ( 𝐽 × 𝐾)
25 fndmin 7065 . . . 4 (((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) Fn ( 𝐽 × 𝐾) ∧ (2nd ↾ ( 𝐽 × 𝐾)) Fn ( 𝐽 × 𝐾)) → dom ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) ∩ (2nd ↾ ( 𝐽 × 𝐾))) = {𝑎 ∈ ( 𝐽 × 𝐾) ∣ ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = ((2nd ↾ ( 𝐽 × 𝐾))‘𝑎)})
2621, 24, 25sylancl 586 . . 3 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) ∩ (2nd ↾ ( 𝐽 × 𝐾))) = {𝑎 ∈ ( 𝐽 × 𝐾) ∣ ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = ((2nd ↾ ( 𝐽 × 𝐾))‘𝑎)})
27 fgraphxp 43216 . . . 4 (𝐹: 𝐽 𝐾𝐹 = {𝑎 ∈ ( 𝐽 × 𝐾) ∣ (𝐹‘(1st𝑎)) = (2nd𝑎)})
2818, 27syl 17 . . 3 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 = {𝑎 ∈ ( 𝐽 × 𝐾) ∣ (𝐹‘(1st𝑎)) = (2nd𝑎)})
2914, 26, 283eqtr4rd 2788 . 2 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 = dom ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) ∩ (2nd ↾ ( 𝐽 × 𝐾))))
30 simpl 482 . . 3 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Haus)
31 cntop1 23248 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
3231adantl 481 . . . . . 6 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
3315toptopon 22923 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
3432, 33sylib 218 . . . . 5 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘ 𝐽))
35 haustop 23339 . . . . . . 7 (𝐾 ∈ Haus → 𝐾 ∈ Top)
3630, 35syl 17 . . . . . 6 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3716toptopon 22923 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3836, 37sylib 218 . . . . 5 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘ 𝐾))
39 tx1cn 23617 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘ 𝐾)) → (1st ↾ ( 𝐽 × 𝐾)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
4034, 38, 39syl2anc 584 . . . 4 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (1st ↾ ( 𝐽 × 𝐾)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
41 cnco 23274 . . . 4 (((1st ↾ ( 𝐽 × 𝐾)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
4240, 41sylancom 588 . . 3 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
43 tx2cn 23618 . . . 4 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘ 𝐾)) → (2nd ↾ ( 𝐽 × 𝐾)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
4434, 38, 43syl2anc 584 . . 3 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (2nd ↾ ( 𝐽 × 𝐾)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
4530, 42, 44hauseqlcld 23654 . 2 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) ∩ (2nd ↾ ( 𝐽 × 𝐾))) ∈ (Clsd‘(𝐽 ×t 𝐾)))
4629, 45eqeltrd 2841 1 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (Clsd‘(𝐽 ×t 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  cin 3950   cuni 4907   × cxp 5683  dom cdm 5685  cres 5687  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Topctop 22899  TopOnctopon 22916  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: (None)
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