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Theorem hausgraph 38292
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 7357 . . . . . . . . 9 (1st ↾ ( 𝐽 × 𝐾)):( 𝐽 × 𝐾)⟶ 𝐽
2 ffn 6206 . . . . . . . . 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 6435 . . . . . . . 8 (((1st ↾ ( 𝐽 × 𝐾)) Fn ( 𝐽 × 𝐾) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = (𝐹‘((1st ↾ ( 𝐽 × 𝐾))‘𝑎)))
53, 4mpan 708 . . . . . . 7 (𝑎 ∈ ( 𝐽 × 𝐾) → ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = (𝐹‘((1st ↾ ( 𝐽 × 𝐾))‘𝑎)))
65adantl 473 . . . . . 6 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = (𝐹‘((1st ↾ ( 𝐽 × 𝐾))‘𝑎)))
7 fvres 6368 . . . . . . . 8 (𝑎 ∈ ( 𝐽 × 𝐾) → ((1st ↾ ( 𝐽 × 𝐾))‘𝑎) = (1st𝑎))
87fveq2d 6356 . . . . . . 7 (𝑎 ∈ ( 𝐽 × 𝐾) → (𝐹‘((1st ↾ ( 𝐽 × 𝐾))‘𝑎)) = (𝐹‘(1st𝑎)))
98adantl 473 . . . . . 6 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → (𝐹‘((1st ↾ ( 𝐽 × 𝐾))‘𝑎)) = (𝐹‘(1st𝑎)))
106, 9eqtrd 2794 . . . . 5 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = (𝐹‘(1st𝑎)))
11 fvres 6368 . . . . . 6 (𝑎 ∈ ( 𝐽 × 𝐾) → ((2nd ↾ ( 𝐽 × 𝐾))‘𝑎) = (2nd𝑎))
1211adantl 473 . . . . 5 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → ((2nd ↾ ( 𝐽 × 𝐾))‘𝑎) = (2nd𝑎))
1310, 12eqeq12d 2775 . . . 4 (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ ( 𝐽 × 𝐾)) → (((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾)))‘𝑎) = ((2nd ↾ ( 𝐽 × 𝐾))‘𝑎) ↔ (𝐹‘(1st𝑎)) = (2nd𝑎)))
1413rabbidva 3328 . . 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 2760 . . . . . . . 8 𝐽 = 𝐽
16 eqid 2760 . . . . . . . 8 𝐾 = 𝐾
1715, 16cnf 21252 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
1817adantl 473 . . . . . 6 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹: 𝐽 𝐾)
19 fco 6219 . . . . . 6 ((𝐹: 𝐽 𝐾 ∧ (1st ↾ ( 𝐽 × 𝐾)):( 𝐽 × 𝐾)⟶ 𝐽) → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))):( 𝐽 × 𝐾)⟶ 𝐾)
2018, 1, 19sylancl 697 . . . . 5 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))):( 𝐽 × 𝐾)⟶ 𝐾)
21 ffn 6206 . . . . 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 7358 . . . . 5 (2nd ↾ ( 𝐽 × 𝐾)):( 𝐽 × 𝐾)⟶ 𝐾
24 ffn 6206 . . . . 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 6487 . . . 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 697 . . 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 38291 . . . 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 2805 . 2 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 = dom ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) ∩ (2nd ↾ ( 𝐽 × 𝐾))))
31 simpl 474 . . 3 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Haus)
32 cntop1 21246 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
3332adantl 473 . . . . . 6 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
3415toptopon 20924 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
3533, 34sylib 208 . . . . 5 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘ 𝐽))
36 haustop 21337 . . . . . . 7 (𝐾 ∈ Haus → 𝐾 ∈ Top)
3731, 36syl 17 . . . . . 6 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3816toptopon 20924 . . . . . 6 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
3937, 38sylib 208 . . . . 5 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘ 𝐾))
40 tx1cn 21614 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘ 𝐾)) → (1st ↾ ( 𝐽 × 𝐾)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
4135, 39, 40syl2anc 696 . . . 4 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (1st ↾ ( 𝐽 × 𝐾)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
42 cnco 21272 . . . 4 (((1st ↾ ( 𝐽 × 𝐾)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
4341, 42sylancom 704 . . 3 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
44 tx2cn 21615 . . . 4 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘ 𝐾)) → (2nd ↾ ( 𝐽 × 𝐾)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
4535, 39, 44syl2anc 696 . . 3 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (2nd ↾ ( 𝐽 × 𝐾)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
4631, 43, 45hauseqlcld 21651 . 2 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom ((𝐹 ∘ (1st ↾ ( 𝐽 × 𝐾))) ∩ (2nd ↾ ( 𝐽 × 𝐾))) ∈ (Clsd‘(𝐽 ×t 𝐾)))
4730, 46eqeltrd 2839 1 ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (Clsd‘(𝐽 ×t 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {crab 3054  cin 3714   cuni 4588   × cxp 5264  dom cdm 5266  cres 5268  ccom 5270   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  Topctop 20900  TopOnctopon 20917  Clsdccld 21022   Cn ccn 21230  Hauscha 21314   ×t ctx 21565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-topgen 16306  df-top 20901  df-topon 20918  df-bases 20952  df-cld 21025  df-cn 21233  df-haus 21321  df-tx 21567
This theorem is referenced by: (None)
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