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 23772

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 23754	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		n0 4294	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3		hmeocn 23738	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
4	3	adantl 481	. . . . . . 7 ⊢ ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
5		cntop2 23219	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Top)
7		simpll 767	. . . . . . . . 9 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝐽 ∈ Nrm)
8	4	adantr 480	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑓 ∈ (𝐽 Cn 𝐾))
9		simprl 771	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑥 ∈ 𝐾)
10		cnima 23243	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (^◡𝑓 “ 𝑥) ∈ 𝐽)
11	8, 9, 10	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → (^◡𝑓 “ 𝑥) ∈ 𝐽)
12		simprr 773	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))
13	12	elin1d 4145	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦 ∈ (Clsd‘𝐾))
14		cnclima 23246	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (^◡𝑓 “ 𝑦) ∈ (Clsd‘𝐽))
15	8, 13, 14	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → (^◡𝑓 “ 𝑦) ∈ (Clsd‘𝐽))
16	12	elin2d 4146	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦 ∈ 𝒫 𝑥)
17	16	elpwid 4551	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦 ⊆ 𝑥)
18		imass2 6062	. . . . . . . . . 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 23333	. . . . . . . . 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 1375	. . . . . . . 8 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → ∃𝑤 ∈ 𝐽 ((^◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))
22		simpllr 776	. . . . . . . . . 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 771	. . . . . . . . . 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 23743	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 ∈ 𝐽) → (𝑓 “ 𝑤) ∈ 𝐾)
25	22, 23, 24	syl2anc 585	. . . . . . . . 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 781	. . . . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
28		eqid 2737	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐾 = ∪ 𝐾
29	27, 28	hmeof1o 23742	. . . . . . . . . . . . 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 6777	. . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 23007	. . . . . . . . . . . . 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 6782	. . . . . . . . . . . . 13 ⊢ (𝑓:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝑓:∪ 𝐽–onto→∪ 𝐾)
37		forn 6750	. . . . . . . . . . . . 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 3960	. . . . . . . . . . 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 6575	. . . . . . . . . . 11 ⊢ ((Fun 𝑓 ∧ 𝑦 ⊆ ran 𝑓) → ((^◡𝑓 “ 𝑦) ⊆ 𝑤 → 𝑦 ⊆ (𝑓 “ 𝑤)))
41	32, 39, 40	syl2anc 585	. . . . . . . . . 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 4882	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
44	43	ad2antrl 729	. . . . . . . . . . 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 23746	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐾)‘(𝑓 “ 𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
46	22, 44, 45	syl2anc 585	. . . . . . . . . 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 782	. . . . . . . . . . 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 23314	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
49	48	ad3antrrr 731	. . . . . . . . . . . . . 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 23037	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑤) ⊆ ∪ 𝐽)
51	49, 44, 50	syl2anc 585	. . . . . . . . . . . . 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 6779	. . . . . . . . . . . . . 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 3960	. . . . . . . . . . . 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 7001	. . . . . . . . . . . 12 ⊢ ((Fun 𝑓 ∧ ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))
56	32, 54, 55	syl2anc 585	. . . . . . . . . . 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 257	. . . . . . . . . 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 3957	. . . . . . . . 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 3949	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑓 “ 𝑤) → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ (𝑓 “ 𝑤)))
60		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑓 “ 𝑤) → ((cls‘𝐾)‘𝑧) = ((cls‘𝐾)‘(𝑓 “ 𝑤)))
61	60	sseq1d 3954	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑓 “ 𝑤) → (((cls‘𝐾)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐾)‘(𝑓 “ 𝑤)) ⊆ 𝑥))
62	59, 61	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑧 = (𝑓 “ 𝑤) → ((𝑦 ⊆ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥) ↔ (𝑦 ⊆ (𝑓 “ 𝑤) ∧ ((cls‘𝐾)‘(𝑓 “ 𝑤)) ⊆ 𝑥)))
63	62	rspcev 3565	. . . . . . . . 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 837	. . . . . . . 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 3145	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → ∃𝑧 ∈ 𝐾 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
66	65	ralrimivva 3181	. . . . . 6 ⊢ ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝐾 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
67		isnrm 23313	. . . . . 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 584	. . . . 5 ⊢ ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Nrm)
69	68	expcom 413	. . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))
70	69	exlimiv 1932	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))
71	2, 70	sylbi 217	. 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))
72	1, 71	sylbi 217	1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 ∩ cin 3889 ⊆ wss 3890 ∅c0 4274 𝒫 cpw 4542 ∪ cuni 4851 class class class wbr 5086 ^◡ccnv 5624 dom cdm 5625 ran crn 5626 “ cima 5628 Fun wfun 6487 –onto→wfo 6491 –1-1-onto→wf1o 6492 ‘cfv 6493 (class class class)co 7361 Topctop 22871 Clsdccld 22994 clsccl 22996 Cn ccn 23202 Nrmcnrm 23288 Homeochmeo 23731 ≃ chmph 23732

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-rep 5213 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-reu 3344 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-int 4891 df-iun 4936 df-iin 4937 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-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7936 df-2nd 7937 df-1o 8399 df-map 8769 df-top 22872 df-topon 22889 df-cld 22997 df-cls 22999 df-cn 23205 df-nrm 23295 df-hmeo 23733 df-hmph 23734

This theorem is referenced by: (None)

W3C validator