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Theorem reghmph 21406

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 21389	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		n0 3890	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3		hmeocn 21373	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
4	3	adantl 481	. . . . . . 7 ⊢ ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
5		cntop2 20855	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Top)
7		simpll 786	. . . . . . . . 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 790	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝐾)
10		cnima 20879	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (^◡𝑓 “ 𝑥) ∈ 𝐽)
11	8, 9, 10	syl2anc 691	. . . . . . . . 9 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → (^◡𝑓 “ 𝑥) ∈ 𝐽)
12		eqid 2610	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
13		eqid 2610	. . . . . . . . . . . . 13 ⊢ ∪ 𝐾 = ∪ 𝐾
14	12, 13	hmeof1o 21377	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:∪ 𝐽–1-1-onto→∪ 𝐾)
15	14	ad2antlr 759	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑓:∪ 𝐽–1-1-onto→∪ 𝐾)
16		f1ocnv 6062	. . . . . . . . . . 11 ⊢ (𝑓:∪ 𝐽–1-1-onto→∪ 𝐾 → ^◡𝑓:∪ 𝐾–1-1-onto→∪ 𝐽)
17		f1ofn 6051	. . . . . . . . . . 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 4403	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐾 → 𝑥 ⊆ ∪ 𝐾)
20	19	ad2antrl 760	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ⊆ ∪ 𝐾)
21		simprr 792	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑥)
22		fnfvima 6400	. . . . . . . . . 10 ⊢ ((^◡𝑓 Fn ∪ 𝐾 ∧ 𝑥 ⊆ ∪ 𝐾 ∧ 𝑦 ∈ 𝑥) → (^◡𝑓‘𝑦) ∈ (^◡𝑓 “ 𝑥))
23	18, 20, 21, 22	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → (^◡𝑓‘𝑦) ∈ (^◡𝑓 “ 𝑥))
24		regsep 20948	. . . . . . . . 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 1318	. . . . . . . 8 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → ∃𝑤 ∈ 𝐽 ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))
26		simpllr 795	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑓 ∈ (𝐽Homeo𝐾))
27		simprl 790	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑤 ∈ 𝐽)
28		hmeoima 21378	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 ∈ 𝐽) → (𝑓 “ 𝑤) ∈ 𝐾)
29	26, 27, 28	syl2anc 691	. . . . . . . . 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 3569	. . . . . . . . . . . 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 800	. . . . . . . . . . 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 6245	. . . . . . . . . . . 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 959	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑦 ∈ (^◡^◡𝑓 “ 𝑤))
37		imacnvcnv 5517	. . . . . . . . . 10 ⊢ (^◡^◡𝑓 “ 𝑤) = (𝑓 “ 𝑤)
38	36, 37	syl6eleq 2698	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑦 ∈ (𝑓 “ 𝑤))
39		elssuni 4403	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
40	39	ad2antrl 760	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑤 ⊆ ∪ 𝐽)
41	12	hmeocls 21381	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐾)‘(𝑓 “ 𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
42	26, 40, 41	syl2anc 691	. . . . . . . . . 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 801	. . . . . . . . . . 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 6052	. . . . . . . . . . . . 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 20947	. . . . . . . . . . . . . . 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 20673	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑤) ⊆ ∪ 𝐽)
51	49, 40, 50	syl2anc 691	. . . . . . . . . . . . 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 6054	. . . . . . . . . . . . . 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	sseqtr4d 3605	. . . . . . . . . . . 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 6241	. . . . . . . . . . . 12 ⊢ ((Fun 𝑓 ∧ ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))
56	46, 54, 55	syl2anc 691	. . . . . . . . . . 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 246	. . . . . . . . . 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 3602	. . . . . . . . 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 2677	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑓 “ 𝑤) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ (𝑓 “ 𝑤)))
60		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑓 “ 𝑤) → ((cls‘𝐾)‘𝑧) = ((cls‘𝐾)‘(𝑓 “ 𝑤)))
61	60	sseq1d 3595	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑓 “ 𝑤) → (((cls‘𝐾)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐾)‘(𝑓 “ 𝑤)) ⊆ 𝑥))
62	59, 61	anbi12d 743	. . . . . . . . . 10 ⊢ (𝑧 = (𝑓 “ 𝑤) → ((𝑦 ∈ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥) ↔ (𝑦 ∈ (𝑓 “ 𝑤) ∧ ((cls‘𝐾)‘(𝑓 “ 𝑤)) ⊆ 𝑥)))
63	62	rspcev 3282	. . . . . . . . 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 1316	. . . . . . . 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 3017	. . . . . . 7 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → ∃𝑧 ∈ 𝐾 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
66	65	ralrimivva 2954	. . . . . 6 ⊢ ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐾 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
67		isreg 20946	. . . . . 6 ⊢ (𝐾 ∈ Reg ↔ (𝐾 ∈ Top ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐾 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥)))
68	6, 66, 67	sylanbrc 695	. . . . 5 ⊢ ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Reg)
69	68	expcom 450	. . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
70	69	exlimiv 1845	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
71	2, 70	sylbi 206	. 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Reg → 𝐾 ∈ Reg))
72	1, 71	sylbi 206	1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Reg → 𝐾 ∈ Reg))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 ∅c0 3874 ∪ cuni 4372 class class class wbr 4583 ^◡ccnv 5037 dom cdm 5038 “ cima 5041 Fun wfun 5798 Fn wfn 5799 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 Topctop 20517 clsccl 20632 Cn ccn 20838 Regcreg 20923 Homeochmeo 21366 ≃ chmph 21367

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-1o 7447 df-map 7746 df-top 20521 df-topon 20523 df-cld 20633 df-cls 20635 df-cn 20841 df-reg 20930 df-hmeo 21368 df-hmph 21369

This theorem is referenced by: (None)

W3C validator