nrmhmph - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > nrmhmph		Structured version Visualization version GIF version

Theorem nrmhmph 23851

Description: Normality is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)

**Assertion**
Ref	Expression
nrmhmph	⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))

Proof of Theorem nrmhmph Dummy variables 𝑤 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hmph 23833	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		n0 4305	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3		hmeocn 23817	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
4	3	adantl 485	. . . . . . 7 ⊢ ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
5		cntop2 23298	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Top)
7		simpll 776	. . . . . . . . 9 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝐽 ∈ Nrm)
8	4	adantr 484	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑓 ∈ (𝐽 Cn 𝐾))
9		simprl 780	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑥 ∈ 𝐾)
10		cnima 23322	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (^◡𝑓 “ 𝑥) ∈ 𝐽)
11	8, 9, 10	syl2anc 593	. . . . . . . . 9 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → (^◡𝑓 “ 𝑥) ∈ 𝐽)
12		simprr 782	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))
13	12	elin1d 4156	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦 ∈ (Clsd‘𝐾))
14		cnclima 23325	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (^◡𝑓 “ 𝑦) ∈ (Clsd‘𝐽))
15	8, 13, 14	syl2anc 593	. . . . . . . . 9 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → (^◡𝑓 “ 𝑦) ∈ (Clsd‘𝐽))
16	12	elin2d 4157	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦 ∈ 𝒫 𝑥)
17	16	elpwid 4564	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦 ⊆ 𝑥)
18		imass2 6091	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝑥 → (^◡𝑓 “ 𝑦) ⊆ (^◡𝑓 “ 𝑥))
19	17, 18	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → (^◡𝑓 “ 𝑦) ⊆ (^◡𝑓 “ 𝑥))
20		nrmsep3 23412	. . . . . . . . 9 ⊢ ((𝐽 ∈ Nrm ∧ ((^◡𝑓 “ 𝑥) ∈ 𝐽 ∧ (^◡𝑓 “ 𝑦) ∈ (Clsd‘𝐽) ∧ (^◡𝑓 “ 𝑦) ⊆ (^◡𝑓 “ 𝑥))) → ∃𝑤 ∈ 𝐽 ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))
21	7, 11, 15, 19, 20	syl13anc 1391	. . . . . . . 8 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → ∃𝑤 ∈ 𝐽 ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))
22		simpllr 785	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑓 ∈ (𝐽Homeo𝐾))
23		simprl 780	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑤 ∈ 𝐽)
24		hmeoima 23822	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 ∈ 𝐽) → (𝑓 “ 𝑤) ∈ 𝐾)
25	22, 23, 24	syl2anc 593	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → (𝑓 “ 𝑤) ∈ 𝐾)
26		simprrl 790	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → (^◡𝑓 “ 𝑦) ⊆ 𝑤)
27		eqid 2762	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
28		eqid 2762	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐾 = ∪ 𝐾
29	27, 28	hmeof1o 23821	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:∪ 𝐽–1-1-onto→∪ 𝐾)
30	22, 29	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑓:∪ 𝐽–1-1-onto→∪ 𝐾)
31		f1ofun 6808	. . . . . . . . . . . 12 ⊢ (𝑓:∪ 𝐽–1-1-onto→∪ 𝐾 → Fun 𝑓)
32	30, 31	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → Fun 𝑓)
33	13	adantr 484	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑦 ∈ (Clsd‘𝐾))
34	28	cldss 23086	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (Clsd‘𝐾) → 𝑦 ⊆ ∪ 𝐾)
35	33, 34	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑦 ⊆ ∪ 𝐾)
36		f1ofo 6814	. . . . . . . . . . . . 13 ⊢ (𝑓:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝑓:∪ 𝐽–onto→∪ 𝐾)
37		forn 6781	. . . . . . . . . . . . 13 ⊢ (𝑓:∪ 𝐽–onto→∪ 𝐾 → ran 𝑓 = ∪ 𝐾)
38	30, 36, 37	3syl 18	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ran 𝑓 = ∪ 𝐾)
39	35, 38	sseqtrrd 3973	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑦 ⊆ ran 𝑓)
40		funimass1 6603	. . . . . . . . . . 11 ⊢ ((Fun 𝑓 ∧ 𝑦 ⊆ ran 𝑓) → ((^◡𝑓 “ 𝑦) ⊆ 𝑤 → 𝑦 ⊆ (𝑓 “ 𝑤)))
41	32, 39, 40	syl2anc 593	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ((^◡𝑓 “ 𝑦) ⊆ 𝑤 → 𝑦 ⊆ (𝑓 “ 𝑤)))
42	26, 41	mpd 15	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑦 ⊆ (𝑓 “ 𝑤))
43		elssuni 4897	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
44	43	ad2antrl 738	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑤 ⊆ ∪ 𝐽)
45	27	hmeocls 23825	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐾)‘(𝑓 “ 𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
46	22, 44, 45	syl2anc 593	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ((cls‘𝐾)‘(𝑓 “ 𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
47		simprrr 791	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥))
48		nrmtop 23393	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
49	48	ad3antrrr 740	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝐽 ∈ Top)
50	27	clsss3 23116	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑤) ⊆ ∪ 𝐽)
51	49, 44, 50	syl2anc 593	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ ∪ 𝐽)
52		f1odm 6810	. . . . . . . . . . . . . 14 ⊢ (𝑓:∪ 𝐽–1-1-onto→∪ 𝐾 → dom 𝑓 = ∪ 𝐽)
53	30, 52	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → dom 𝑓 = ∪ 𝐽)
54	51, 53	sseqtrrd 3973	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓)
55		funimass3 7035	. . . . . . . . . . . 12 ⊢ ((Fun 𝑓 ∧ ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))
56	32, 54, 55	syl2anc 593	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))
57	47, 56	mpbird 259	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → (𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥)
58	46, 57	eqsstrd 3970	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ((cls‘𝐾)‘(𝑓 “ 𝑤)) ⊆ 𝑥)
59		sseq2 3962	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑓 “ 𝑤) → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ (𝑓 “ 𝑤)))
60		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑓 “ 𝑤) → ((cls‘𝐾)‘𝑧) = ((cls‘𝐾)‘(𝑓 “ 𝑤)))
61	60	sseq1d 3967	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑓 “ 𝑤) → (((cls‘𝐾)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐾)‘(𝑓 “ 𝑤)) ⊆ 𝑥))
62	59, 61	anbi12d 641	. . . . . . . . . 10 ⊢ (𝑧 = (𝑓 “ 𝑤) → ((𝑦 ⊆ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥) ↔ (𝑦 ⊆ (𝑓 “ 𝑤) ∧ ((cls‘𝐾)‘(𝑓 “ 𝑤)) ⊆ 𝑥)))
63	62	rspcev 3581	. . . . . . . . 9 ⊢ (((𝑓 “ 𝑤) ∈ 𝐾 ∧ (𝑦 ⊆ (𝑓 “ 𝑤) ∧ ((cls‘𝐾)‘(𝑓 “ 𝑤)) ⊆ 𝑥)) → ∃𝑧 ∈ 𝐾 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
64	25, 42, 58, 63	syl12anc 847	. . . . . . . 8 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ∃𝑧 ∈ 𝐾 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
65	21, 64	rexlimddv 3169	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → ∃𝑧 ∈ 𝐾 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
66	65	ralrimivva 3205	. . . . . 6 ⊢ ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝐾 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
67		isnrm 23392	. . . . . 6 ⊢ (𝐾 ∈ Nrm ↔ (𝐾 ∈ Top ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝐾 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥)))
68	6, 66, 67	sylanbrc 592	. . . . 5 ⊢ ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Nrm)
69	68	expcom 417	. . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))
70	69	exlimiv 1950	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))
71	2, 70	sylbi 219	. 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))
72	1, 71	sylbi 219	1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1560 ∃wex 1799 ∈ wcel 2142 ≠ wne 2957 ∀wral 3076 ∃wrex 3086 ∩ cin 3903 ⊆ wss 3904 ∅c0 4285 𝒫 cpw 4555 ∪ cuni 4865 class class class wbr 5100 ^◡ccnv 5646 dom cdm 5647 ran crn 5648 “ cima 5650 Fun wfun 6515 –onto→wfo 6519 –1-1-onto→wf1o 6520 ‘cfv 6521 (class class class)co 7396 Topctop 22950 Clsdccld 23073 clsccl 23075 Cn ccn 23281 Nrmcnrm 23367 Homeochmeo 23810 ≃ chmph 23811

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-rep 5227 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-reu 3368 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-int 4906 df-iun 4951 df-iin 4952 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-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-ov 7399 df-oprab 7400 df-mpo 7401 df-1st 7970 df-2nd 7971 df-1o 8437 df-map 8810 df-top 22951 df-topon 22968 df-cld 23076 df-cls 23078 df-cn 23284 df-nrm 23374 df-hmeo 23812 df-hmph 23813

This theorem is referenced by: (None)

W3C validator