Theorem nrmhmph 24168

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 24150	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		n0 4303	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3		hmeocn 24134	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
4	3	adantl 487	. . . . . . 7 ⊢ ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
5		cntop2 23614	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
6	4, 5	syl 18	. . . . . 6 ⊢ ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Top)
7		simpll 779	. . . . . . . . 9 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝐽 ∈ Nrm)
8	4	adantr 486	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑓 ∈ (𝐽 Cn 𝐾))
9		simprl 783	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑥 ∈ 𝐾)
10		cnima 23638	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (◡𝑓 “ 𝑥) ∈ 𝐽)
11	8, 9, 10	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → (◡𝑓 “ 𝑥) ∈ 𝐽)
12		simprr 785	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))
13	12	elin1d 4104	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦 ∈ (Clsd‘𝐾))
14		cnclima 23641	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (◡𝑓 “ 𝑦) ∈ (Clsd‘𝐽))
15	8, 13, 14	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → (◡𝑓 “ 𝑦) ∈ (Clsd‘𝐽))
16	12	elin2d 4105	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦 ∈ 𝒫 𝑥)
17	16	elpwid 4569	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦 ⊆ 𝑥)
18		imass2 6073	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝑥 → (◡𝑓 “ 𝑦) ⊆ (◡𝑓 “ 𝑥))
19	17, 18	syl 18	. . . . . . . . 9 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → (◡𝑓 “ 𝑦) ⊆ (◡𝑓 “ 𝑥))
20		nrmsep3 23728	. . . . . . . . 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 1400	. . . . . . . 8 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → ∃𝑤 ∈ 𝐽 ((◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝑓 “ 𝑥)))
22		simpllr 788	. . . . . . . . . 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 783	. . . . . . . . . 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 24139	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 ∈ 𝐽) → (𝑓 “ 𝑤) ∈ 𝐾)
25	22, 23, 24	syl2anc 596	. . . . . . . . 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 793	. . . . . . . . . 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 2763	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
28		eqid 2763	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐾 = ∪ 𝐾
29	27, 28	hmeof1o 24138	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:∪ 𝐽⟶1-1onto∪ 𝐾)
30	22, 29	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝑓 “ 𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝑓 “ 𝑥)))) → 𝑓:∪ 𝐽⟶1-1onto∪ 𝐾)
31		f1ofun 6856	. . . . . . . . . . . 12 ⊢ (𝑓:∪ 𝐽⟶1-1onto∪ 𝐾 → Fun 𝑓)
32	30, 31	syl 18	. . . . . . . . . . 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 486	. . . . . . . . . . . . 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 23402	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (Clsd‘𝐾) → 𝑦 ⊆ ∪ 𝐾)
35	33, 34	syl 18	. . . . . . . . . . . 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 6862	. . . . . . . . . . . . 13 ⊢ (𝑓:∪ 𝐽⟶1-1onto∪ 𝐾 → 𝑓:∪ 𝐽⟶onto∪ 𝐾)
37		forn 6829	. . . . . . . . . . . . 13 ⊢ (𝑓:∪ 𝐽⟶onto∪ 𝐾 → ran 𝑓 = ∪ 𝐾)
38	30, 36, 37	3syl 19	. . . . . . . . . . . 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 3950	. . . . . . . . . . 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 6652	. . . . . . . . . . 11 ⊢ ((Fun 𝑓 ∧ 𝑦 ⊆ ran 𝑓) → ((◡𝑓 “ 𝑦) ⊆ 𝑤 → 𝑦 ⊆ (𝑓 “ 𝑤)))
41	32, 39, 40	syl2anc 596	. . . . . . . . . 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 16	. . . . . . . . 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 4904	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
44	43	ad2antrl 741	. . . . . . . . . . 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 24142	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐾)‘(𝑓 “ 𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
46	22, 44, 45	syl2anc 596	. . . . . . . . . 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 794	. . . . . . . . . . 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 23709	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top)
49	48	ad3antrrr 743	. . . . . . . . . . . . . 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 23432	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑤) ⊆ ∪ 𝐽)
51	49, 44, 50	syl2anc 596	. . . . . . . . . . . . 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 6858	. . . . . . . . . . . . . 14 ⊢ (𝑓:∪ 𝐽⟶1-1onto∪ 𝐾 → dom 𝑓 = ∪ 𝐽)
53	30, 52	syl 18	. . . . . . . . . . . . 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 3950	. . . . . . . . . . . 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 7084	. . . . . . . . . . . 12 ⊢ ((Fun 𝑓 ∧ ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (◡𝑓 “ 𝑥)))
56	32, 54, 55	syl2anc 596	. . . . . . . . . . 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 260	. . . . . . . . . 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 3947	. . . . . . . . 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 3939	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑓 “ 𝑤) → (𝑦 ⊆ 𝑧 ↔ 𝑦 ⊆ (𝑓 “ 𝑤)))
60		fveq2 6916	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑓 “ 𝑤) → ((cls‘𝐾)‘𝑧) = ((cls‘𝐾)‘(𝑓 “ 𝑤)))
61	60	sseq1d 3944	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑓 “ 𝑤) → (((cls‘𝐾)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐾)‘(𝑓 “ 𝑤)) ⊆ 𝑥))
62	59, 61	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑧 = (𝑓 “ 𝑤) → ((𝑦 ⊆ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥) ↔ (𝑦 ⊆ (𝑓 “ 𝑤) ∧ ((cls‘𝐾)‘(𝑓 “ 𝑤)) ⊆ 𝑥)))
63	62	rspcev 3580	. . . . . . . . 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 850	. . . . . . . 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 3173	. . . . . . 7 ⊢ (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → ∃𝑧 ∈ 𝐾 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
66	65	ralrimivva 3209	. . . . . 6 ⊢ ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝐾 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
67		isnrm 23708	. . . . . 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 595	. . . . 5 ⊢ ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Nrm)
69	68	expcom 419	. . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))
70	69	exlimiv 1965	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))
71	2, 70	sylbi 220	. 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))
72	1, 71	sylbi 220	1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  𝒫 cpw 4560  ∪ cuni 4872   class class class wbr 5108  ^◡ccnv 5657  dom cdm 5658  ran crn 5659   “ cima 5661  Fun wfun 6560  ⟶ontowfo 6564  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451  Topctop 23266  Clsdccld 23389  clsccl 23391   Cn ccn 23597  Nrmcnrm 23683  Homeochmeo 24127   ≃ chmph 24128

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-1o 8514  df-map 8887  df-top 23267  df-topon 23284  df-cld 23392  df-cls 23394  df-cn 23600  df-nrm 23690  df-hmeo 24129  df-hmph 24130

This theorem is used by: (None)