hausgraph - Mathbox for Stefan O'Rear

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Theorem hausgraph 43779

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 7994	. . . . . . . . 9 ⊢ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐽
2		ffn 6691	. . . . . . . . 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 6964	. . . . . . . 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 700	. . . . . . 7 ⊢ (𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) → ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)))‘𝑎) = (𝐹‘((1^st ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎)))
6	5	adantl 485	. . . . . 6 ⊢ (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ (∪ 𝐽 × ∪ 𝐾)) → ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)))‘𝑎) = (𝐹‘((1^st ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎)))
7		fvres 6886	. . . . . . . 8 ⊢ (𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) → ((1^st ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎) = (1^st ‘𝑎))
8	7	fveq2d 6871	. . . . . . 7 ⊢ (𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) → (𝐹‘((1^st ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎)) = (𝐹‘(1^st ‘𝑎)))
9	8	adantl 485	. . . . . 6 ⊢ (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ (∪ 𝐽 × ∪ 𝐾)) → (𝐹‘((1^st ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎)) = (𝐹‘(1^st ‘𝑎)))
10	6, 9	eqtrd 2797	. . . . 5 ⊢ (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ (∪ 𝐽 × ∪ 𝐾)) → ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)))‘𝑎) = (𝐹‘(1^st ‘𝑎)))
11		fvres 6886	. . . . . 6 ⊢ (𝑎 ∈ (∪ 𝐽 × ∪ 𝐾) → ((2^nd ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎) = (2^nd ‘𝑎))
12	11	adantl 485	. . . . 5 ⊢ (((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑎 ∈ (∪ 𝐽 × ∪ 𝐾)) → ((2^nd ↾ (∪ 𝐽 × ∪ 𝐾))‘𝑎) = (2^nd ‘𝑎))
13	10, 12	eqeq12d 2778	. . . 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 3420	. . 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 2762	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
16		eqid 2762	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
17	15, 16	cnf 23303	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
18	17	adantl 485	. . . . . 6 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
19		fco 6716	. . . . . 6 ⊢ ((𝐹:∪ 𝐽⟶∪ 𝐾 ∧ (1^st ↾ (∪ 𝐽 × ∪ 𝐾)):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐽) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐾)
20	18, 1, 19	sylancl 595	. . . . 5 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐾)
21	20	ffnd 6692	. . . 4 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))) Fn (∪ 𝐽 × ∪ 𝐾))
22		f2ndres 7995	. . . . 5 ⊢ (2^nd ↾ (∪ 𝐽 × ∪ 𝐾)):(∪ 𝐽 × ∪ 𝐾)⟶∪ 𝐾
23		ffn 6691	. . . . 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 7026	. . . 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 595	. . 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 43778	. . . 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 2808	. 2 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 = dom ((𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))) ∩ (2^nd ↾ (∪ 𝐽 × ∪ 𝐾))))
30		simpl 486	. . 3 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Haus)
31		cntop1 23297	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
32	31	adantl 485	. . . . . 6 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
33	15	toptopon 22974	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
34	32, 33	sylib 220	. . . . 5 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
35		haustop 23388	. . . . . . 7 ⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top)
36	30, 35	syl 17	. . . . . 6 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
37	16	toptopon 22974	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
38	36, 37	sylib 220	. . . . 5 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
39		tx1cn 23666	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → (1^st ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
40	34, 38, 39	syl2anc 593	. . . 4 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (1^st ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
41		cnco 23323	. . . 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 597	. . 3 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ (1^st ↾ (∪ 𝐽 × ∪ 𝐾))) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
43		tx2cn 23667	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾)) → (2^nd ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
44	34, 38, 43	syl2anc 593	. . 3 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (2^nd ↾ (∪ 𝐽 × ∪ 𝐾)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
45	30, 42, 44	hauseqlcld 23703	. 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 2862	1 ⊢ ((𝐾 ∈ Haus ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (Clsd‘(𝐽 ×_t 𝐾)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1560 ∈ wcel 2142 {crab 3414 ∩ cin 3903 ∪ cuni 4865 × cxp 5645 dom cdm 5647 ↾ cres 5649 ∘ ccom 5651 Fn wfn 6516 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 1^st c1st 7968 2^nd c2nd 7969 Topctop 22950 TopOnctopon 22967 Clsdccld 23073 Cn ccn 23281 Hauscha 23365 ×_t ctx 23617

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-fv 6529 df-ov 7399 df-oprab 7400 df-mpo 7401 df-1st 7970 df-2nd 7971 df-map 8810 df-topgen 17472 df-top 22951 df-topon 22968 df-bases 23003 df-cld 23076 df-cn 23284 df-haus 23372 df-tx 23619

This theorem is referenced by: (None)

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