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Theorem hausgraph 37310
 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 7150 . . . . . . . . 9 (1st ↾ ( 𝐽 × 𝐾)):( 𝐽 × 𝐾)⟶ 𝐽
2 ffn 6012 . . . . . . . . 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 6240 . . . . . . . 8 (((1st ↾ ( 𝐽 × 𝐾)) Fn ( 𝐽 × 𝐾) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = (𝐹‘((1st ↾ ( 𝐽 × 𝐾))‘𝑎)))
53, 4mpan 705 . . . . . . 7 (𝑎 ∈ ( 𝐽 × 𝐾) → ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = (𝐹‘((1st ↾ ( 𝐽 × 𝐾))‘𝑎)))
65adantl 482 . . . . . 6 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = (𝐹‘((1st ↾ ( 𝐽 × 𝐾))‘𝑎)))
7 fvres 6174 . . . . . . . 8 (𝑎 ∈ ( 𝐽 × 𝐾) → ((1st ↾ ( 𝐽 × 𝐾))‘𝑎) = (1st𝑎))
87fveq2d 6162 . . . . . . 7 (𝑎 ∈ ( 𝐽 × 𝐾) → (𝐹‘((1st ↾ ( 𝐽 × 𝐾))‘𝑎)) = (𝐹‘(1st𝑎)))
98adantl 482 . . . . . 6 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → (𝐹‘((1st ↾ ( 𝐽 × 𝐾))‘𝑎)) = (𝐹‘(1st𝑎)))
106, 9eqtrd 2655 . . . . 5 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = (𝐹‘(1st𝑎)))
11 fvres 6174 . . . . . 6 (𝑎 ∈ ( 𝐽 × 𝐾) → ((2nd ↾ ( 𝐽 × 𝐾))‘𝑎) = (2nd𝑎))
1211adantl 482 . . . . 5 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → ((2nd ↾ ( 𝐽 × 𝐾))‘𝑎) = (2nd𝑎))
1310, 12eqeq12d 2636 . . . 4 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → (((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = ((2nd ↾ ( 𝐽 × 𝐾))‘𝑎) ↔ (𝐹‘(1st𝑎)) = (2nd𝑎)))
1413rabbidva 3180 . . 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 2621 . . . . . . . 8 𝐽 = 𝐽
16 eqid 2621 . . . . . . . 8 𝐾 = 𝐾
1715, 16cnf 20990 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
1817adantl 482 . . . . . 6 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽 𝐾)
19 fco 6025 . . . . . 6 ((𝐹: 𝐽 𝐾 ∧ (1st ↾ ( 𝐽 × 𝐾)):( 𝐽 × 𝐾)⟶ 𝐽) → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))):( 𝐽 × 𝐾)⟶ 𝐾)
2018, 1, 19sylancl 693 . . . . 5 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))):( 𝐽 × 𝐾)⟶ 𝐾)
21 ffn 6012 . . . . 5 ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))):( 𝐽 × 𝐾)⟶ 𝐾 → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) Fn ( 𝐽 × 𝐾))
2220, 21syl 17 . . . 4 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) Fn ( 𝐽 × 𝐾))
23 f2ndres 7151 . . . . 5 (2nd ↾ ( 𝐽 × 𝐾)):( 𝐽 × 𝐾)⟶ 𝐾
24 ffn 6012 . . . . 5 ((2nd ↾ ( 𝐽 × 𝐾)):( 𝐽 × 𝐾)⟶ 𝐾 → (2nd ↾ ( 𝐽 × 𝐾)) Fn ( 𝐽 × 𝐾))
2523, 24ax-mp 5 . . . 4 (2nd ↾ ( 𝐽 × 𝐾)) Fn ( 𝐽 × 𝐾)
26 fndmin 6290 . . . 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)})
2722, 25, 26sylancl 693 . . 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)})
28 fgraphxp 37309 . . . 4 (𝐹: 𝐽 𝐾𝐹 = {𝑎 ∈ ( 𝐽 × 𝐾) ∣ (𝐹‘(1st𝑎)) = (2nd𝑎)})
2918, 28syl 17 . . 3 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 = {𝑎 ∈ ( 𝐽 × 𝐾) ∣ (𝐹‘(1st𝑎)) = (2nd𝑎)})
3014, 27, 293eqtr4rd 2666 . 2 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 = dom ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) ∩ (2nd ↾ ( 𝐽 × 𝐾))))
31 simpl 473 . . 3 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Haus)
32 cntop1 20984 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
3332adantl 482 . . . . . 6 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
3415toptopon 20662 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
3533, 34sylib 208 . . . . 5 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘ 𝐽))
36 haustop 21075 . . . . . . 7 (𝐾 ∈ Haus → 𝐾 ∈ Top)
3731, 36syl 17 . . . . . 6 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3816toptopon 20662 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3937, 38sylib 208 . . . . 5 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘ 𝐾))
40 tx1cn 21352 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘ 𝐾)) → (1st ↾ ( 𝐽 × 𝐾)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
4135, 39, 40syl2anc 692 . . . 4 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (1st ↾ ( 𝐽 × 𝐾)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
42 cnco 21010 . . . 4 (((1st ↾ ( 𝐽 × 𝐾)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
4341, 42sylancom 700 . . 3 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
44 tx2cn 21353 . . . 4 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘ 𝐾)) → (2nd ↾ ( 𝐽 × 𝐾)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
4535, 39, 44syl2anc 692 . . 3 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (2nd ↾ ( 𝐽 × 𝐾)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
4631, 43, 45hauseqlcld 21389 . 2 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) ∩ (2nd ↾ ( 𝐽 × 𝐾))) ∈ (Clsd‘(𝐽 ×t 𝐾)))
4730, 46eqeltrd 2698 1 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (Clsd‘(𝐽 ×t 𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  {crab 2912   ∩ cin 3559  ∪ cuni 4409   × cxp 5082  dom cdm 5084   ↾ cres 5086   ∘ ccom 5088   Fn wfn 5852  ⟶wf 5853  ‘cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  Topctop 20638  TopOnctopon 20655  Clsdccld 20760   Cn ccn 20968  Hauscha 21052   ×t ctx 21303 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 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-map 7819  df-topgen 16044  df-top 20639  df-topon 20656  df-bases 20690  df-cld 20763  df-cn 20971  df-haus 21059  df-tx 21305 This theorem is referenced by: (None)
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