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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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