reghmph - Metamath Proof Explorer

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

Theorem reghmph 23833

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 23816	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		n0 4305	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3		hmeocn 23800	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
4	3	adantl 485	. . . . . . 7 ⊢ ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
5		cntop2 23281	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Top)
7		simpll 776	. . . . . . . . 9 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Reg)
8	4	adantr 484	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑓 ∈ (𝐽 Cn 𝐾))
9		simprl 780	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝐾)
10		cnima 23305	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (^◡𝑓 “ 𝑥) ∈ 𝐽)
11	8, 9, 10	syl2anc 593	. . . . . . . . 9 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → (^◡𝑓 “ 𝑥) ∈ 𝐽)
12		eqid 2761	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
13		eqid 2761	. . . . . . . . . . . . 13 ⊢ ∪ 𝐾 = ∪ 𝐾
14	12, 13	hmeof1o 23804	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:∪ 𝐽–1-1-onto→∪ 𝐾)
15	14	ad2antlr 737	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑓:∪ 𝐽–1-1-onto→∪ 𝐾)
16		f1ocnv 6815	. . . . . . . . . . 11 ⊢ (𝑓:∪ 𝐽–1-1-onto→∪ 𝐾 → ^◡𝑓:∪ 𝐾–1-1-onto→∪ 𝐽)
17		f1ofn 6803	. . . . . . . . . . 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 4896	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐾 → 𝑥 ⊆ ∪ 𝐾)
20	19	ad2antrl 738	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ⊆ ∪ 𝐾)
21		simprr 782	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑥)
22		fnfvima 7213	. . . . . . . . . 10 ⊢ ((^◡𝑓 Fn ∪ 𝐾 ∧ 𝑥 ⊆ ∪ 𝐾 ∧ 𝑦 ∈ 𝑥) → (^◡𝑓‘𝑦) ∈ (^◡𝑓 “ 𝑥))
23	18, 20, 21, 22	syl3anc 1389	. . . . . . . . 9 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → (^◡𝑓‘𝑦) ∈ (^◡𝑓 “ 𝑥))
24		regsep 23374	. . . . . . . . 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 1389	. . . . . . . 8 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → ∃𝑤 ∈ 𝐽 ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))
26		simpllr 785	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑓 ∈ (𝐽Homeo𝐾))
27		simprl 780	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑤 ∈ 𝐽)
28		hmeoima 23805	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 ∈ 𝐽) → (𝑓 “ 𝑤) ∈ 𝐾)
29	26, 27, 28	syl2anc 593	. . . . . . . . 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 3937	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ ∪ 𝐾)
31	30	adantr 484	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑦 ∈ ∪ 𝐾)
32		simprrl 790	. . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → ^◡𝑓 Fn ∪ 𝐾)
34		elpreima 7035	. . . . . . . . . . . 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 723	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑦 ∈ (^◡^◡𝑓 “ 𝑤))
37		imacnvcnv 6189	. . . . . . . . . 10 ⊢ (^◡^◡𝑓 “ 𝑤) = (𝑓 “ 𝑤)
38	36, 37	eleqtrdi 2871	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑦 ∈ (𝑓 “ 𝑤))
39		elssuni 4896	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
40	39	ad2antrl 738	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑤 ⊆ ∪ 𝐽)
41	12	hmeocls 23808	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐾)‘(𝑓 “ 𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
42	26, 40, 41	syl2anc 593	. . . . . . . . . 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 791	. . . . . . . . . . 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 484	. . . . . . . . . . . . 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 6804	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝐽 ∈ Reg)
48		regtop 23373	. . . . . . . . . . . . . . 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 23099	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑤) ⊆ ∪ 𝐽)
51	49, 40, 50	syl2anc 593	. . . . . . . . . . . . 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 6806	. . . . . . . . . . . . . 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 3973	. . . . . . . . . . . 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 7031	. . . . . . . . . . . 12 ⊢ ((Fun 𝑓 ∧ ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))
56	46, 54, 55	syl2anc 593	. . . . . . . . . . 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 259	. . . . . . . . . 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 3970	. . . . . . . . 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 2850	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑓 “ 𝑤) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ (𝑓 “ 𝑤)))
60		fveq2 6863	. . . . . . . . . . . 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	29, 38, 58, 63	syl12anc 847	. . . . . . . 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 3168	. . . . . . 7 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → ∃𝑧 ∈ 𝐾 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
66	65	ralrimivva 3204	. . . . . 6 ⊢ ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐾 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
67		isreg 23372	. . . . . 6 ⊢ (𝐾 ∈ Reg ↔ (𝐾 ∈ Top ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐾 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥)))
68	6, 66, 67	sylanbrc 592	. . . . 5 ⊢ ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Reg)
69	68	expcom 417	. . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
70	69	exlimiv 1949	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
71	2, 70	sylbi 219	. 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
72	1, 71	sylbi 219	1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Reg → 𝐾 ∈ Reg))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∃wex 1798 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 ∃wrex 3085 ⊆ wss 3904 ∅c0 4285 ∪ cuni 4864 class class class wbr 5099 ^◡ccnv 5644 dom cdm 5645 “ cima 5648 Fun wfun 6511 Fn wfn 6512 –1-1-onto→wf1o 6516 ‘cfv 6517 (class class class)co 7392 Topctop 22933 clsccl 23058 Cn ccn 23264 Regcreg 23349 Homeochmeo 23793 ≃ chmph 23794

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-ov 7395 df-oprab 7396 df-mpo 7397 df-1st 7966 df-2nd 7967 df-1o 8432 df-map 8805 df-top 22934 df-topon 22951 df-cld 23059 df-cls 23061 df-cn 23267 df-reg 23356 df-hmeo 23795 df-hmph 23796

This theorem is referenced by: (None)

W3C validator