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Theorem reghmph 23680

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

**Assertion**
Ref	Expression
reghmph	⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Reg → 𝐾 ∈ Reg))

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

Step	Hyp	Ref	Expression
1		hmph 23663	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		n0 4316	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3		hmeocn 23647	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
4	3	adantl 481	. . . . . . 7 ⊢ ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
5		cntop2 23128	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Top)
7		simpll 766	. . . . . . . . 9 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Reg)
8	4	adantr 480	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑓 ∈ (𝐽 Cn 𝐾))
9		simprl 770	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝐾)
10		cnima 23152	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (^◡𝑓 “ 𝑥) ∈ 𝐽)
11	8, 9, 10	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → (^◡𝑓 “ 𝑥) ∈ 𝐽)
12		eqid 2729	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
13		eqid 2729	. . . . . . . . . . . . 13 ⊢ ∪ 𝐾 = ∪ 𝐾
14	12, 13	hmeof1o 23651	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:∪ 𝐽–1-1-onto→∪ 𝐾)
15	14	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑓:∪ 𝐽–1-1-onto→∪ 𝐾)
16		f1ocnv 6812	. . . . . . . . . . 11 ⊢ (𝑓:∪ 𝐽–1-1-onto→∪ 𝐾 → ^◡𝑓:∪ 𝐾–1-1-onto→∪ 𝐽)
17		f1ofn 6801	. . . . . . . . . . 11 ⊢ (^◡𝑓:∪ 𝐾–1-1-onto→∪ 𝐽 → ^◡𝑓 Fn ∪ 𝐾)
18	15, 16, 17	3syl 18	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → ^◡𝑓 Fn ∪ 𝐾)
19		elssuni 4901	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐾 → 𝑥 ⊆ ∪ 𝐾)
20	19	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ⊆ ∪ 𝐾)
21		simprr 772	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑥)
22		fnfvima 7207	. . . . . . . . . 10 ⊢ ((^◡𝑓 Fn ∪ 𝐾 ∧ 𝑥 ⊆ ∪ 𝐾 ∧ 𝑦 ∈ 𝑥) → (^◡𝑓‘𝑦) ∈ (^◡𝑓 “ 𝑥))
23	18, 20, 21, 22	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → (^◡𝑓‘𝑦) ∈ (^◡𝑓 “ 𝑥))
24		regsep 23221	. . . . . . . . 9 ⊢ ((𝐽 ∈ Reg ∧ (^◡𝑓 “ 𝑥) ∈ 𝐽 ∧ (^◡𝑓‘𝑦) ∈ (^◡𝑓 “ 𝑥)) → ∃𝑤 ∈ 𝐽 ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))
25	7, 11, 23, 24	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → ∃𝑤 ∈ 𝐽 ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))
26		simpllr 775	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑓 ∈ (𝐽Homeo𝐾))
27		simprl 770	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑤 ∈ 𝐽)
28		hmeoima 23652	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 ∈ 𝐽) → (𝑓 “ 𝑤) ∈ 𝐾)
29	26, 27, 28	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → (𝑓 “ 𝑤) ∈ 𝐾)
30	20, 21	sseldd 3947	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ ∪ 𝐾)
31	30	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑦 ∈ ∪ 𝐾)
32		simprrl 780	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → (^◡𝑓‘𝑦) ∈ 𝑤)
33	18	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ^◡𝑓 Fn ∪ 𝐾)
34		elpreima 7030	. . . . . . . . . . . 12 ⊢ (^◡𝑓 Fn ∪ 𝐾 → (𝑦 ∈ (^◡^◡𝑓 “ 𝑤) ↔ (𝑦 ∈ ∪ 𝐾 ∧ (^◡𝑓‘𝑦) ∈ 𝑤)))
35	33, 34	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → (𝑦 ∈ (^◡^◡𝑓 “ 𝑤) ↔ (𝑦 ∈ ∪ 𝐾 ∧ (^◡𝑓‘𝑦) ∈ 𝑤)))
36	31, 32, 35	mpbir2and 713	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑦 ∈ (^◡^◡𝑓 “ 𝑤))
37		imacnvcnv 6179	. . . . . . . . . 10 ⊢ (^◡^◡𝑓 “ 𝑤) = (𝑓 “ 𝑤)
38	36, 37	eleqtrdi 2838	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑦 ∈ (𝑓 “ 𝑤))
39		elssuni 4901	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
40	39	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑤 ⊆ ∪ 𝐽)
41	12	hmeocls 23655	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐾)‘(𝑓 “ 𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
42	26, 40, 41	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ((cls‘𝐾)‘(𝑓 “ 𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
43		simprrr 781	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥))
44	15	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑓:∪ 𝐽–1-1-onto→∪ 𝐾)
45		f1ofun 6802	. . . . . . . . . . . . 13 ⊢ (𝑓:∪ 𝐽–1-1-onto→∪ 𝐾 → Fun 𝑓)
46	44, 45	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → Fun 𝑓)
47	7	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝐽 ∈ Reg)
48		regtop 23220	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Reg → 𝐽 ∈ Top)
49	47, 48	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝐽 ∈ Top)
50	12	clsss3 22946	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑤) ⊆ ∪ 𝐽)
51	49, 40, 50	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ ∪ 𝐽)
52		f1odm 6804	. . . . . . . . . . . . . 14 ⊢ (𝑓:∪ 𝐽–1-1-onto→∪ 𝐾 → dom 𝑓 = ∪ 𝐽)
53	44, 52	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → dom 𝑓 = ∪ 𝐽)
54	51, 53	sseqtrrd 3984	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓)
55		funimass3 7026	. . . . . . . . . . . 12 ⊢ ((Fun 𝑓 ∧ ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))
56	46, 54, 55	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))
57	43, 56	mpbird 257	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → (𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥)
58	42, 57	eqsstrd 3981	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ((cls‘𝐾)‘(𝑓 “ 𝑤)) ⊆ 𝑥)
59		eleq2 2817	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑓 “ 𝑤) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ (𝑓 “ 𝑤)))
60		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑓 “ 𝑤) → ((cls‘𝐾)‘𝑧) = ((cls‘𝐾)‘(𝑓 “ 𝑤)))
61	60	sseq1d 3978	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑓 “ 𝑤) → (((cls‘𝐾)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐾)‘(𝑓 “ 𝑤)) ⊆ 𝑥))
62	59, 61	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑧 = (𝑓 “ 𝑤) → ((𝑦 ∈ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥) ↔ (𝑦 ∈ (𝑓 “ 𝑤) ∧ ((cls‘𝐾)‘(𝑓 “ 𝑤)) ⊆ 𝑥)))
63	62	rspcev 3588	. . . . . . . . 9 ⊢ (((𝑓 “ 𝑤) ∈ 𝐾 ∧ (𝑦 ∈ (𝑓 “ 𝑤) ∧ ((cls‘𝐾)‘(𝑓 “ 𝑤)) ⊆ 𝑥)) → ∃𝑧 ∈ 𝐾 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
64	29, 38, 58, 63	syl12anc 836	. . . . . . . 8 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ∃𝑧 ∈ 𝐾 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
65	25, 64	rexlimddv 3140	. . . . . . 7 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → ∃𝑧 ∈ 𝐾 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
66	65	ralrimivva 3180	. . . . . 6 ⊢ ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐾 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
67		isreg 23219	. . . . . 6 ⊢ (𝐾 ∈ Reg ↔ (𝐾 ∈ Top ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐾 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥)))
68	6, 66, 67	sylanbrc 583	. . . . 5 ⊢ ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Reg)
69	68	expcom 413	. . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
70	69	exlimiv 1930	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
71	2, 70	sylbi 217	. 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
72	1, 71	sylbi 217	1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Reg → 𝐾 ∈ Reg))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ⊆ wss 3914 ∅c0 4296 ∪ cuni 4871 class class class wbr 5107 ^◡ccnv 5637 dom cdm 5638 “ cima 5641 Fun wfun 6505 Fn wfn 6506 –1-1-onto→wf1o 6510 ‘cfv 6511 (class class class)co 7387 Topctop 22780 clsccl 22905 Cn ccn 23111 Regcreg 23196 Homeochmeo 23640 ≃ chmph 23641

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-1st 7968 df-2nd 7969 df-1o 8434 df-map 8801 df-top 22781 df-topon 22798 df-cld 22906 df-cls 22908 df-cn 23114 df-reg 23203 df-hmeo 23642 df-hmph 23643

This theorem is referenced by: (None)

W3C validator