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Theorem hausgraph 43194
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 8037 . . . . . . . . 9 (1st ↾ ( 𝐽 × 𝐾)):( 𝐽 × 𝐾)⟶ 𝐽
2 ffn 6737 . . . . . . . . 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 6926 . . . . . . . 8 (𝑎 ∈ ( 𝐽 × 𝐾) → ((1st ↾ ( 𝐽 × 𝐾))‘𝑎) = (1st𝑎))
87fveq2d 6911 . . . . . . 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 2775 . . . . 5 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = (𝐹‘(1st𝑎)))
11 fvres 6926 . . . . . 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 2751 . . . 4 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → (((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = ((2nd ↾ ( 𝐽 × 𝐾))‘𝑎) ↔ (𝐹‘(1st𝑎)) = (2nd𝑎)))
1413rabbidva 3440 . . 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 2735 . . . . . . . 8 𝐽 = 𝐽
16 eqid 2735 . . . . . . . 8 𝐾 = 𝐾
1715, 16cnf 23270 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
1817adantl 481 . . . . . 6 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽 𝐾)
19 fco 6761 . . . . . 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 6738 . . . 4 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) Fn ( 𝐽 × 𝐾))
22 f2ndres 8038 . . . . 5 (2nd ↾ ( 𝐽 × 𝐾)):( 𝐽 × 𝐾)⟶ 𝐾
23 ffn 6737 . . . . 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 43193 . . . 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 2786 . 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 23264 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
3231adantl 481 . . . . . 6 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
3315toptopon 22939 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
3432, 33sylib 218 . . . . 5 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘ 𝐽))
35 haustop 23355 . . . . . . 7 (𝐾 ∈ Haus → 𝐾 ∈ Top)
3630, 35syl 17 . . . . . 6 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3716toptopon 22939 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3836, 37sylib 218 . . . . 5 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘ 𝐾))
39 tx1cn 23633 . . . . 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 23290 . . . 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 23634 . . . 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 23670 . 2 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) ∩ (2nd ↾ ( 𝐽 × 𝐾))) ∈ (Clsd‘(𝐽 ×t 𝐾)))
4629, 45eqeltrd 2839 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 1537  wcel 2106  {crab 3433  cin 3962   cuni 4912   × cxp 5687  dom cdm 5689  cres 5691  ccom 5693   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  Topctop 22915  TopOnctopon 22932  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: (None)
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