hausgraph - Mathbox for Stefan O'Rear

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Theorem hausgraph 43178

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 7948	. . . . . . . . 9 ⊢ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐽
2		ffn 6652	. . . . . . . . 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 6920	. . . . . . . 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 690	. . . . . . 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 6841	. . . . . . . 8 ⊢ (𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) → ((1^st ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎) = (1^st ‘𝑎))
8	7	fveq2d 6826	. . . . . . 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 2764	. . . . 5 ⊢ (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ (∪ 𝐽 × ∪ 𝐾)) → ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)))‘𝑎) = (𝐹‘(1^st ‘𝑎)))
11		fvres 6841	. . . . . 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 2745	. . . 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 3401	. . 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 2729	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
16		eqid 2729	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
17	15, 16	cnf 23131	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
18	17	adantl 481	. . . . . 6 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
19		fco 6676	. . . . . 6 ⊢ ((𝐹:∪ 𝐽⟶∪ 𝐾 ∧ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐽) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐾)
20	18, 1, 19	sylancl 586	. . . . 5 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐾)
21	20	ffnd 6653	. . . 4 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))) Fn (∪ 𝐽 × ∪ 𝐾))
22		f2ndres 7949	. . . . 5 ⊢ (2^nd ↾ (∪ 𝐽 × ∪ 𝐾)):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐾
23		ffn 6652	. . . . 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 6979	. . . 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 586	. . 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 43177	. . . 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 2775	. 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 23125	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
32	31	adantl 481	. . . . . 6 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
33	15	toptopon 22802	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
34	32, 33	sylib 218	. . . . 5 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
35		haustop 23216	. . . . . . 7 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
36	30, 35	syl 17	. . . . . 6 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
37	16	toptopon 22802	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
38	36, 37	sylib 218	. . . . 5 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
39		tx1cn 23494	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → (1^st ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
40	34, 38, 39	syl2anc 584	. . . 4 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (1^st ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
41		cnco 23151	. . . 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 588	. . 3 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
43		tx2cn 23495	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → (2^nd ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
44	34, 38, 43	syl2anc 584	. . 3 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (2^nd ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
45	30, 42, 44	hauseqlcld 23531	. 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 2828	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 1540 ∈ wcel 2109 {crab 3394 ∩ cin 3902 ∪ cuni 4858 × cxp 5617 dom cdm 5619 ↾ cres 5621 ∘ ccom 5623 Fn wfn 6477 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 1^st c1st 7922 2^nd c2nd 7923 Topctop 22778 TopOnctopon 22795 Clsdccld 22901 Cn ccn 23109 Hauscha 23193 ×_t ctx 23445

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-1st 7924 df-2nd 7925 df-map 8755 df-topgen 17347 df-top 22779 df-topon 22796 df-bases 22831 df-cld 22904 df-cn 23112 df-haus 23200 df-tx 23447

This theorem is referenced by: (None)

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