hausgraph - Mathbox for Stefan O'Rear

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Theorem hausgraph 43654

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 7960	. . . . . . . . 9 ⊢ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐽
2		ffn 6663	. . . . . . . . 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 6932	. . . . . . . 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 691	. . . . . . 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 6854	. . . . . . . 8 ⊢ (𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) → ((1^st ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎) = (1^st ‘𝑎))
8	7	fveq2d 6839	. . . . . . 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 2772	. . . . 5 ⊢ (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ (∪ 𝐽 × ∪ 𝐾)) → ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)))‘𝑎) = (𝐹‘(1^st ‘𝑎)))
11		fvres 6854	. . . . . 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 2753	. . . 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 3396	. . 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 2737	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
16		eqid 2737	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
17	15, 16	cnf 23224	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
18	17	adantl 481	. . . . . 6 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
19		fco 6687	. . . . . 6 ⊢ ((𝐹:∪ 𝐽⟶∪ 𝐾 ∧ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐽) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐾)
20	18, 1, 19	sylancl 587	. . . . 5 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐾)
21	20	ffnd 6664	. . . 4 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))) Fn (∪ 𝐽 × ∪ 𝐾))
22		f2ndres 7961	. . . . 5 ⊢ (2^nd ↾ (∪ 𝐽 × ∪ 𝐾)):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐾
23		ffn 6663	. . . . 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 6992	. . . 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 587	. . 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 43653	. . . 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 2783	. 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 23218	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
32	31	adantl 481	. . . . . 6 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
33	15	toptopon 22895	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
34	32, 33	sylib 218	. . . . 5 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
35		haustop 23309	. . . . . . 7 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
36	30, 35	syl 17	. . . . . 6 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
37	16	toptopon 22895	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
38	36, 37	sylib 218	. . . . 5 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
39		tx1cn 23587	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → (1^st ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
40	34, 38, 39	syl2anc 585	. . . 4 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (1^st ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
41		cnco 23244	. . . 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 589	. . 3 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
43		tx2cn 23588	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → (2^nd ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
44	34, 38, 43	syl2anc 585	. . 3 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (2^nd ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
45	30, 42, 44	hauseqlcld 23624	. 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 2837	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 1542 ∈ wcel 2114 {crab 3390 ∩ cin 3889 ∪ cuni 4851 × cxp 5623 dom cdm 5625 ↾ cres 5627 ∘ ccom 5629 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 1^st c1st 7934 2^nd c2nd 7935 Topctop 22871 TopOnctopon 22888 Clsdccld 22994 Cn ccn 23202 Hauscha 23286 ×_t ctx 23538

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7936 df-2nd 7937 df-map 8769 df-topgen 17400 df-top 22872 df-topon 22889 df-bases 22924 df-cld 22997 df-cn 23205 df-haus 23293 df-tx 23540

This theorem is referenced by: (None)

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