hausgraph - Mathbox for Stefan O'Rear

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Theorem hausgraph 43166

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.

Step	Hyp	Ref	Expression
1		f1stres 8054	. . . . . . . . 9 ⊢ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐽
2		ffn 6747	. . . . . . . . 9 ⊢ ((1^st ↾ (∪ 𝐽 × ∪ 𝐾)):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐽 → (1^st ↾ (∪ 𝐽 × ∪ 𝐾)) Fn (∪ 𝐽 × ∪ 𝐾))
3	1, 2	ax-mp 5	. . . . . . . 8 ⊢ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)) Fn (∪ 𝐽 × ∪ 𝐾)
4		fvco2 7019	. . . . . . . 8 ⊢ (((1^st ↾ (∪ 𝐽 × ∪ 𝐾)) Fn (∪ 𝐽 × ∪ 𝐾) ∧ 𝑎 ∈ (∪ 𝐽 × ∪ 𝐾)) → ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)))‘𝑎) = (𝐹‘((1^st ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎)))
5	3, 4	mpan 689	. . . . . . 7 ⊢ (𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) → ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)))‘𝑎) = (𝐹‘((1^st ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎)))
6	5	adantl 481	. . . . . 6 ⊢ (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ (∪ 𝐽 × ∪ 𝐾)) → ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)))‘𝑎) = (𝐹‘((1^st ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎)))
7		fvres 6939	. . . . . . . 8 ⊢ (𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) → ((1^st ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎) = (1^st ‘𝑎))
8	7	fveq2d 6924	. . . . . . 7 ⊢ (𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) → (𝐹‘((1^st ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎)) = (𝐹‘(1^st ‘𝑎)))
9	8	adantl 481	. . . . . 6 ⊢ (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ (∪ 𝐽 × ∪ 𝐾)) → (𝐹‘((1^st ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎)) = (𝐹‘(1^st ‘𝑎)))
10	6, 9	eqtrd 2780	. . . . 5 ⊢ (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ (∪ 𝐽 × ∪ 𝐾)) → ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)))‘𝑎) = (𝐹‘(1^st ‘𝑎)))
11		fvres 6939	. . . . . 6 ⊢ (𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) → ((2^nd ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎) = (2^nd ‘𝑎))
12	11	adantl 481	. . . . 5 ⊢ (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ (∪ 𝐽 × ∪ 𝐾)) → ((2^nd ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎) = (2^nd ‘𝑎))
13	10, 12	eqeq12d 2756	. . . 4 ⊢ (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ (∪ 𝐽 × ∪ 𝐾)) → (((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)))‘𝑎) = ((2^nd ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎) ↔ (𝐹‘(1^st ‘𝑎)) = (2^nd ‘𝑎)))
14	13	rabbidva 3450	. . 3 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → {𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) ∣ ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)))‘𝑎) = ((2^nd ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎)} = {𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) ∣ (𝐹‘(1^st ‘𝑎)) = (2^nd ‘𝑎)})
15		eqid 2740	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
16		eqid 2740	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
17	15, 16	cnf 23275	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
18	17	adantl 481	. . . . . 6 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
19		fco 6771	. . . . . 6 ⊢ ((𝐹:∪ 𝐽⟶∪ 𝐾 ∧ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐽) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐾)
20	18, 1, 19	sylancl 585	. . . . 5 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐾)
21	20	ffnd 6748	. . . 4 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))) Fn (∪ 𝐽 × ∪ 𝐾))
22		f2ndres 8055	. . . . 5 ⊢ (2^nd ↾ (∪ 𝐽 × ∪ 𝐾)):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐾
23		ffn 6747	. . . . 5 ⊢ ((2^nd ↾ (∪ 𝐽 × ∪ 𝐾)):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐾 → (2^nd ↾ (∪ 𝐽 × ∪ 𝐾)) Fn (∪ 𝐽 × ∪ 𝐾))
24	22, 23	ax-mp 5	. . . 4 ⊢ (2^nd ↾ (∪ 𝐽 × ∪ 𝐾)) Fn (∪ 𝐽 × ∪ 𝐾)
25		fndmin 7078	. . . 4 ⊢ (((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))) Fn (∪ 𝐽 × ∪ 𝐾) ∧ (2^nd ↾ (∪ 𝐽 × ∪ 𝐾)) Fn (∪ 𝐽 × ∪ 𝐾)) → dom ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))) ∩ (2^nd ↾ (∪ 𝐽 × ∪ 𝐾))) = {𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) ∣ ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)))‘𝑎) = ((2^nd ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎)})
26	21, 24, 25	sylancl 585	. . 3 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))) ∩ (2^nd ↾ (∪ 𝐽 × ∪ 𝐾))) = {𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) ∣ ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)))‘𝑎) = ((2^nd ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎)})
27		fgraphxp 43165	. . . 4 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → 𝐹 = {𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) ∣ (𝐹‘(1^st ‘𝑎)) = (2^nd ‘𝑎)})
28	18, 27	syl 17	. . 3 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 = {𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) ∣ (𝐹‘(1^st ‘𝑎)) = (2^nd ‘𝑎)})
29	14, 26, 28	3eqtr4rd 2791	. 2 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 = dom ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))) ∩ (2^nd ↾ (∪ 𝐽 × ∪ 𝐾))))
30		simpl 482	. . 3 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Haus)
31		cntop1 23269	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
32	31	adantl 481	. . . . . 6 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
33	15	toptopon 22944	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
34	32, 33	sylib 218	. . . . 5 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
35		haustop 23360	. . . . . . 7 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
36	30, 35	syl 17	. . . . . 6 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
37	16	toptopon 22944	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
38	36, 37	sylib 218	. . . . 5 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
39		tx1cn 23638	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → (1^st ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
40	34, 38, 39	syl2anc 583	. . . 4 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (1^st ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
41		cnco 23295	. . . 4 ⊢ (((1^st ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
42	40, 41	sylancom 587	. . 3 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
43		tx2cn 23639	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → (2^nd ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
44	34, 38, 43	syl2anc 583	. . 3 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (2^nd ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
45	30, 42, 44	hauseqlcld 23675	. 2 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → dom ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))) ∩ (2^nd ↾ (∪ 𝐽 × ∪ 𝐾))) ∈ (Clsd‘(𝐽 ×_t 𝐾)))
46	29, 45	eqeltrd 2844	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 2108 {crab 3443 ∩ cin 3975 ∪ cuni 4931 × cxp 5698 dom cdm 5700 ↾ cres 5702 ∘ ccom 5704 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 Topctop 22920 TopOnctopon 22937 Clsdccld 23045 Cn ccn 23253 Hauscha 23337 ×_t ctx 23589

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-map 8886 df-topgen 17503 df-top 22921 df-topon 22938 df-bases 22974 df-cld 23048 df-cn 23256 df-haus 23344 df-tx 23591

This theorem is referenced by: (None)

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