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Theorem reghmph 23817

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 23800	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		n0 4359	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3		hmeocn 23784	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
4	3	adantl 481	. . . . . . 7 ⊢ ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
5		cntop2 23265	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
6	4, 5	syl 17	. . . . . 6 ⊢ ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Top)
7		simpll 767	. . . . . . . . 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 771	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝐾)
10		cnima 23289	. . . . . . . . . 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 2735	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
13		eqid 2735	. . . . . . . . . . . . 13 ⊢ ∪ 𝐾 = ∪ 𝐾
14	12, 13	hmeof1o 23788	. . . . . . . . . . . 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 6861	. . . . . . . . . . 11 ⊢ (𝑓:∪ 𝐽–1-1-onto→∪ 𝐾 → ^◡𝑓:∪ 𝐾–1-1-onto→∪ 𝐽)
17		f1ofn 6850	. . . . . . . . . . 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 4942	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐾 → 𝑥 ⊆ ∪ 𝐾)
20	19	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ⊆ ∪ 𝐾)
21		simprr 773	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑥)
22		fnfvima 7253	. . . . . . . . . 10 ⊢ ((^◡𝑓 Fn ∪ 𝐾 ∧ 𝑥 ⊆ ∪ 𝐾 ∧ 𝑦 ∈ 𝑥) → (^◡𝑓‘𝑦) ∈ (^◡𝑓 “ 𝑥))
23	18, 20, 21, 22	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → (^◡𝑓‘𝑦) ∈ (^◡𝑓 “ 𝑥))
24		regsep 23358	. . . . . . . . 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 1370	. . . . . . . 8 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → ∃𝑤 ∈ 𝐽 ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))
26		simpllr 776	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑓 ∈ (𝐽Homeo𝐾))
27		simprl 771	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑤 ∈ 𝐽)
28		hmeoima 23789	. . . . . . . . . 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 3996	. . . . . . . . . . . 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 781	. . . . . . . . . . 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 7078	. . . . . . . . . . . 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 6228	. . . . . . . . . 10 ⊢ (^◡^◡𝑓 “ 𝑤) = (𝑓 “ 𝑤)
38	36, 37	eleqtrdi 2849	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) ∧ (𝑤 ∈ 𝐽 ∧ ((^◡𝑓‘𝑦) ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (^◡𝑓 “ 𝑥)))) → 𝑦 ∈ (𝑓 “ 𝑤))
39		elssuni 4942	. . . . . . . . . . . 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 23792	. . . . . . . . . . 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 782	. . . . . . . . . . 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 6851	. . . . . . . . . . . . 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 23357	. . . . . . . . . . . . . . 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 23083	. . . . . . . . . . . . . 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 6853	. . . . . . . . . . . . . 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 4037	. . . . . . . . . . . 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 7074	. . . . . . . . . . . 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 4034	. . . . . . . . 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 2828	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑓 “ 𝑤) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ (𝑓 “ 𝑤)))
60		fveq2 6907	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑓 “ 𝑤) → ((cls‘𝐾)‘𝑧) = ((cls‘𝐾)‘(𝑓 “ 𝑤)))
61	60	sseq1d 4027	. . . . . . . . . . 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 3622	. . . . . . . . 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 837	. . . . . . . 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 3159	. . . . . . 7 ⊢ (((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑥)) → ∃𝑧 ∈ 𝐾 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
66	65	ralrimivva 3200	. . . . . 6 ⊢ ((𝐽 ∈ Reg ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐾 (𝑦 ∈ 𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
67		isreg 23356	. . . . . 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 1928	. . 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 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 ∅c0 4339 ∪ cuni 4912 class class class wbr 5148 ^◡ccnv 5688 dom cdm 5689 “ cima 5692 Fun wfun 6557 Fn wfn 6558 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 Topctop 22915 clsccl 23042 Cn ccn 23248 Regcreg 23333 Homeochmeo 23777 ≃ chmph 23778

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-1o 8505 df-map 8867 df-top 22916 df-topon 22933 df-cld 23043 df-cls 23045 df-cn 23251 df-reg 23340 df-hmeo 23779 df-hmph 23780

This theorem is referenced by: (None)

W3C validator