Theorem symgtgp 23609

Description: The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.) (Proof shortened by AV, 30-Mar-2024.)

Hypothesis

Ref	Expression
symgtgp.g	⊢ 𝐺 = (SymGrp‘𝐴)

Assertion

Ref	Expression
symgtgp	⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ TopGrp)

Proof of Theorem symgtgp

Dummy variables 𝑡 𝑓 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		symgtgp.g	. . 3 ⊢ 𝐺 = (SymGrp‘𝐴)
2	1	symggrp 19267	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Grp)
3		eqid 2732	. . . 4 ⊢ (EndoFMnd‘𝐴) = (EndoFMnd‘𝐴)
4	3	efmndtmd 23604	. . 3 ⊢ (𝐴 ∈ 𝑉 → (EndoFMnd‘𝐴) ∈ TopMnd)
5		eqid 2732	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	3, 1, 5	symgsubmefmnd 19265	. . 3 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝐺) ∈ (SubMnd‘(EndoFMnd‘𝐴)))
7	1, 5, 3	symgressbas 19248	. . . 4 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s (Base‘𝐺))
8	7	submtmd 23607	. . 3 ⊢ (((EndoFMnd‘𝐴) ∈ TopMnd ∧ (Base‘𝐺) ∈ (SubMnd‘(EndoFMnd‘𝐴))) → 𝐺 ∈ TopMnd)
9	4, 6, 8	syl2anc 584	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ TopMnd)
10		eqid 2732	. . . . . 6 ⊢ (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
11	1, 5	symgtopn 19273	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) = (TopOpen‘𝐺))
12		distopon 22499	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
13	10	pttoponconst 23100	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)))
14	12, 13	mpdan 685	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)))
15	1, 5	elsymgbas 19240	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝐺) ↔ 𝑥:𝐴–1-1-onto→𝐴))
16		f1of 6833	. . . . . . . . . . 11 ⊢ (𝑥:𝐴–1-1-onto→𝐴 → 𝑥:𝐴⟶𝐴)
17		elmapg 8832	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ (𝐴 ↑m 𝐴) ↔ 𝑥:𝐴⟶𝐴))
18	17	anidms 567	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (𝐴 ↑m 𝐴) ↔ 𝑥:𝐴⟶𝐴))
19	16, 18	imbitrrid 245	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → (𝑥:𝐴–1-1-onto→𝐴 → 𝑥 ∈ (𝐴 ↑m 𝐴)))
20	15, 19	sylbid 239	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝐺) → 𝑥 ∈ (𝐴 ↑m 𝐴)))
21	20	ssrdv 3988	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝐺) ⊆ (𝐴 ↑m 𝐴))
22		resttopon 22664	. . . . . . . 8 ⊢ (((∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)) ∧ (Base‘𝐺) ⊆ (𝐴 ↑m 𝐴)) → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) ∈ (TopOn‘(Base‘𝐺)))
23	14, 21, 22	syl2anc 584	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) ∈ (TopOn‘(Base‘𝐺)))
24	11, 23	eqeltrrd 2834	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
25		id 22	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
26		distop 22497	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
27		fconst6g 6780	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ Top → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
28	26, 27	syl 17	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
29	15	biimpa 477	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴–1-1-onto→𝐴)
30		f1ocnv 6845	. . . . . . . . . . . 12 ⊢ (𝑥:𝐴–1-1-onto→𝐴 → ◡𝑥:𝐴–1-1-onto→𝐴)
31		f1of 6833	. . . . . . . . . . . 12 ⊢ (◡𝑥:𝐴–1-1-onto→𝐴 → ◡𝑥:𝐴⟶𝐴)
32	29, 30, 31	3syl 18	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → ◡𝑥:𝐴⟶𝐴)
33	32	ffvelcdmda 7086	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ 𝐴) → (◡𝑥‘𝑦) ∈ 𝐴)
34	33	an32s 650	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ (Base‘𝐺)) → (◡𝑥‘𝑦) ∈ 𝐴)
35	34	fmpttd 7114	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)):(Base‘𝐺)⟶𝐴)
36	35	adantr 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)):(Base‘𝐺)⟶𝐴)
37		cnveq 5873	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑓 → ◡𝑥 = ◡𝑓)
38	37	fveq1d 6893	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑓 → (◡𝑥‘𝑦) = (◡𝑓‘𝑦))
39		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) = (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦))
40		fvex 6904	. . . . . . . . . . . . . . 15 ⊢ (◡𝑓‘𝑦) ∈ V
41	38, 39, 40	fvmpt 6998	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (Base‘𝐺) → ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦))‘𝑓) = (◡𝑓‘𝑦))
42	41	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦))‘𝑓) = (◡𝑓‘𝑦))
43	42	eleq1d 2818	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → (((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦))‘𝑓) ∈ 𝑡 ↔ (◡𝑓‘𝑦) ∈ 𝑡))
44		eqid 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(◡𝑓‘𝑦))) = (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(◡𝑓‘𝑦)))
45	44	mptiniseg 6238	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ V → (◡(𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(◡𝑓‘𝑦))) “ {𝑦}) = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦})
46	45	elv 3480	. . . . . . . . . . . . . . . 16 ⊢ (◡(𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(◡𝑓‘𝑦))) “ {𝑦}) = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦}
47		eqid 2732	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺))
48	14	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)))
49	21	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (Base‘𝐺) ⊆ (𝐴 ↑m 𝐴))
50		toponuni 22415	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)) → (𝐴 ↑m 𝐴) = ∪ (∏t‘(𝐴 × {𝒫 𝐴})))
51		mpteq1 5241	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ↑m 𝐴) = ∪ (∏t‘(𝐴 × {𝒫 𝐴})) → (𝑢 ∈ (𝐴 ↑m 𝐴) ↦ (𝑢‘(◡𝑓‘𝑦))) = (𝑢 ∈ ∪ (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(◡𝑓‘𝑦))))
52	48, 50, 51	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (𝐴 ↑m 𝐴) ↦ (𝑢‘(◡𝑓‘𝑦))) = (𝑢 ∈ ∪ (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(◡𝑓‘𝑦))))
53		simpll 765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝐴 ∈ 𝑉)
54	28	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
55	1, 5	elsymgbas 19240	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (Base‘𝐺) ↔ 𝑓:𝐴–1-1-onto→𝐴))
56	55	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → (𝑓 ∈ (Base‘𝐺) ↔ 𝑓:𝐴–1-1-onto→𝐴))
57	56	biimpa 477	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓:𝐴–1-1-onto→𝐴)
58		f1ocnv 6845	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓:𝐴–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→𝐴)
59		f1of 6833	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (◡𝑓:𝐴–1-1-onto→𝐴 → ◡𝑓:𝐴⟶𝐴)
60	57, 58, 59	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ◡𝑓:𝐴⟶𝐴)
61		simplr 767	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑦 ∈ 𝐴)
62	60, 61	ffvelcdmd 7087	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (◡𝑓‘𝑦) ∈ 𝐴)
63		eqid 2732	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∪ (∏t‘(𝐴 × {𝒫 𝐴})) = ∪ (∏t‘(𝐴 × {𝒫 𝐴}))
64	63, 10	ptpjcn 23114	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 × {𝒫 𝐴}):𝐴⟶Top ∧ (◡𝑓‘𝑦) ∈ 𝐴) → (𝑢 ∈ ∪ (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(◡𝑓‘𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn ((𝐴 × {𝒫 𝐴})‘(◡𝑓‘𝑦))))
65	53, 54, 62, 64	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ ∪ (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(◡𝑓‘𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn ((𝐴 × {𝒫 𝐴})‘(◡𝑓‘𝑦))))
66	26	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝒫 𝐴 ∈ Top)
67		fvconst2g 7202	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝒫 𝐴 ∈ Top ∧ (◡𝑓‘𝑦) ∈ 𝐴) → ((𝐴 × {𝒫 𝐴})‘(◡𝑓‘𝑦)) = 𝒫 𝐴)
68	66, 62, 67	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((𝐴 × {𝒫 𝐴})‘(◡𝑓‘𝑦)) = 𝒫 𝐴)
69	68	oveq2d 7424	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((∏t‘(𝐴 × {𝒫 𝐴})) Cn ((𝐴 × {𝒫 𝐴})‘(◡𝑓‘𝑦))) = ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
70	65, 69	eleqtrd 2835	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ ∪ (∏t‘(𝐴 × {𝒫 𝐴})) ↦ (𝑢‘(◡𝑓‘𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
71	52, 70	eqeltrd 2833	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (𝐴 ↑m 𝐴) ↦ (𝑢‘(◡𝑓‘𝑦))) ∈ ((∏t‘(𝐴 × {𝒫 𝐴})) Cn 𝒫 𝐴))
72	47, 48, 49, 71	cnmpt1res 23179	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(◡𝑓‘𝑦))) ∈ (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴))
73	11	oveq1d 7423	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ 𝑉 → (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
74	73	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
75	72, 74	eleqtrd 2835	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(◡𝑓‘𝑦))) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴))
76		snelpwi 5443	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝒫 𝐴)
77	76	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → {𝑦} ∈ 𝒫 𝐴)
78		cnima 22768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(◡𝑓‘𝑦))) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴) ∧ {𝑦} ∈ 𝒫 𝐴) → (◡(𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(◡𝑓‘𝑦))) “ {𝑦}) ∈ (TopOpen‘𝐺))
79	75, 77, 78	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (◡(𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(◡𝑓‘𝑦))) “ {𝑦}) ∈ (TopOpen‘𝐺))
80	46, 79	eqeltrrid 2838	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ∈ (TopOpen‘𝐺))
81	80	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ∈ (TopOpen‘𝐺))
82		fveq1 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑓 → (𝑢‘(◡𝑓‘𝑦)) = (𝑓‘(◡𝑓‘𝑦)))
83	82	eqeq1d 2734	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑓 → ((𝑢‘(◡𝑓‘𝑦)) = 𝑦 ↔ (𝑓‘(◡𝑓‘𝑦)) = 𝑦))
84		simplr 767	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → 𝑓 ∈ (Base‘𝐺))
85	57	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → 𝑓:𝐴–1-1-onto→𝐴)
86		simpllr 774	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → 𝑦 ∈ 𝐴)
87		f1ocnvfv2 7274	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝐴–1-1-onto→𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑓‘(◡𝑓‘𝑦)) = 𝑦)
88	85, 86, 87	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → (𝑓‘(◡𝑓‘𝑦)) = 𝑦)
89	83, 84, 88	elrabd 3685	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → 𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦})
90		ssrab2 4077	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ⊆ (Base‘𝐺)
91	90	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ⊆ (Base‘𝐺))
92	15	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → (𝑥 ∈ (Base‘𝐺) ↔ 𝑥:𝐴–1-1-onto→𝐴))
93	92	biimpa 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴–1-1-onto→𝐴)
94	62	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → (◡𝑓‘𝑦) ∈ 𝐴)
95		f1ocnvfv 7275	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥:𝐴–1-1-onto→𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝐴) → ((𝑥‘(◡𝑓‘𝑦)) = 𝑦 → (◡𝑥‘𝑦) = (◡𝑓‘𝑦)))
96	93, 94, 95	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑥‘(◡𝑓‘𝑦)) = 𝑦 → (◡𝑥‘𝑦) = (◡𝑓‘𝑦)))
97		simplrr 776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → (◡𝑓‘𝑦) ∈ 𝑡)
98		eleq1 2821	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((◡𝑥‘𝑦) = (◡𝑓‘𝑦) → ((◡𝑥‘𝑦) ∈ 𝑡 ↔ (◡𝑓‘𝑦) ∈ 𝑡))
99	97, 98	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((◡𝑥‘𝑦) = (◡𝑓‘𝑦) → (◡𝑥‘𝑦) ∈ 𝑡))
100	96, 99	syld 47	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → ((𝑥‘(◡𝑓‘𝑦)) = 𝑦 → (◡𝑥‘𝑦) ∈ 𝑡))
101	100	ralrimiva 3146	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → ∀𝑥 ∈ (Base‘𝐺)((𝑥‘(◡𝑓‘𝑦)) = 𝑦 → (◡𝑥‘𝑦) ∈ 𝑡))
102		fveq1 6890	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑥 → (𝑢‘(◡𝑓‘𝑦)) = (𝑥‘(◡𝑓‘𝑦)))
103	102	eqeq1d 2734	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑥 → ((𝑢‘(◡𝑓‘𝑦)) = 𝑦 ↔ (𝑥‘(◡𝑓‘𝑦)) = 𝑦))
104	103	ralrab 3689	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑥 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} (◡𝑥‘𝑦) ∈ 𝑡 ↔ ∀𝑥 ∈ (Base‘𝐺)((𝑥‘(◡𝑓‘𝑦)) = 𝑦 → (◡𝑥‘𝑦) ∈ 𝑡))
105	101, 104	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → ∀𝑥 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} (◡𝑥‘𝑦) ∈ 𝑡)
106		ssrab 4070	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ⊆ {𝑥 ∈ (Base‘𝐺) ∣ (◡𝑥‘𝑦) ∈ 𝑡} ↔ ({𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ⊆ (Base‘𝐺) ∧ ∀𝑥 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} (◡𝑥‘𝑦) ∈ 𝑡))
107	91, 105, 106	sylanbrc 583	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ⊆ {𝑥 ∈ (Base‘𝐺) ∣ (◡𝑥‘𝑦) ∈ 𝑡})
108	39	mptpreima 6237	. . . . . . . . . . . . . . . 16 ⊢ (◡(𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ 𝑡) = {𝑥 ∈ (Base‘𝐺) ∣ (◡𝑥‘𝑦) ∈ 𝑡}
109	107, 108	sseqtrrdi 4033	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ⊆ (◡(𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ 𝑡))
110		funmpt 6586	. . . . . . . . . . . . . . . 16 ⊢ Fun (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦))
111		fvex 6904	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑥‘𝑦) ∈ V
112	111, 39	dmmpti 6694	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) = (Base‘𝐺)
113	91, 112	sseqtrrdi 4033	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ⊆ dom (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)))
114		funimass3 7055	. . . . . . . . . . . . . . . 16 ⊢ ((Fun (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∧ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ⊆ dom (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦))) → (((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦}) ⊆ 𝑡 ↔ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ⊆ (◡(𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ 𝑡)))
115	110, 113, 114	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → (((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦}) ⊆ 𝑡 ↔ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ⊆ (◡(𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ 𝑡)))
116	109, 115	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦}) ⊆ 𝑡)
117		eleq2 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} → (𝑓 ∈ 𝑣 ↔ 𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦}))
118		imaeq2 6055	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} → ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ 𝑣) = ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦}))
119	118	sseq1d 4013	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} → (((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ 𝑣) ⊆ 𝑡 ↔ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦}) ⊆ 𝑡))
120	117, 119	anbi12d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} → ((𝑓 ∈ 𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ 𝑣) ⊆ 𝑡) ↔ (𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦}) ⊆ 𝑡)))
121	120	rspcev 3612	. . . . . . . . . . . . . 14 ⊢ (({𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ∈ (TopOpen‘𝐺) ∧ (𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦}) ⊆ 𝑡)) → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓 ∈ 𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ 𝑣) ⊆ 𝑡))
122	81, 89, 116, 121	syl12anc 835	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓 ∈ 𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ 𝑣) ⊆ 𝑡))
123	122	expr 457	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → ((◡𝑓‘𝑦) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓 ∈ 𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ 𝑣) ⊆ 𝑡)))
124	43, 123	sylbid 239	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → (((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦))‘𝑓) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓 ∈ 𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ 𝑣) ⊆ 𝑡)))
125	124	ralrimiva 3146	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ∀𝑡 ∈ 𝒫 𝐴(((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦))‘𝑓) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓 ∈ 𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ 𝑣) ⊆ 𝑡)))
126	24	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
127	12	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝒫 𝐴 ∈ (TopOn‘𝐴))
128		simpr 485	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓 ∈ (Base‘𝐺))
129		iscnp 22740	. . . . . . . . . . 11 ⊢ (((TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)) ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)):(Base‘𝐺)⟶𝐴 ∧ ∀𝑡 ∈ 𝒫 𝐴(((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦))‘𝑓) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓 ∈ 𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ 𝑣) ⊆ 𝑡)))))
130	126, 127, 128, 129	syl3anc 1371	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)):(Base‘𝐺)⟶𝐴 ∧ ∀𝑡 ∈ 𝒫 𝐴(((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦))‘𝑓) ∈ 𝑡 → ∃𝑣 ∈ (TopOpen‘𝐺)(𝑓 ∈ 𝑣 ∧ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ 𝑣) ⊆ 𝑡)))))
131	36, 125, 130	mpbir2and 711	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))
132	131	ralrimiva 3146	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ∀𝑓 ∈ (Base‘𝐺)(𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))
133		cncnp 22783	. . . . . . . . . 10 ⊢ (((TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)) ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)):(Base‘𝐺)⟶𝐴 ∧ ∀𝑓 ∈ (Base‘𝐺)(𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))))
134	24, 12, 133	syl2anc 584	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)):(Base‘𝐺)⟶𝐴 ∧ ∀𝑓 ∈ (Base‘𝐺)(𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))))
135	134	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴) ↔ ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)):(Base‘𝐺)⟶𝐴 ∧ ∀𝑓 ∈ (Base‘𝐺)(𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))))
136	35, 132, 135	mpbir2and 711	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴))
137		fvconst2g 7202	. . . . . . . . 9 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝑦 ∈ 𝐴) → ((𝐴 × {𝒫 𝐴})‘𝑦) = 𝒫 𝐴)
138	26, 137	sylan 580	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ((𝐴 × {𝒫 𝐴})‘𝑦) = 𝒫 𝐴)
139	138	oveq2d 7424	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ((TopOpen‘𝐺) Cn ((𝐴 × {𝒫 𝐴})‘𝑦)) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
140	136, 139	eleqtrrd 2836	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∈ ((TopOpen‘𝐺) Cn ((𝐴 × {𝒫 𝐴})‘𝑦)))
141	10, 24, 25, 28, 140	ptcn 23130	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝐺) ↦ (𝑦 ∈ 𝐴 ↦ (◡𝑥‘𝑦))) ∈ ((TopOpen‘𝐺) Cn (∏t‘(𝐴 × {𝒫 𝐴}))))
142		eqid 2732	. . . . . . . . 9 ⊢ (invg‘𝐺) = (invg‘𝐺)
143	5, 142	grpinvf 18870	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺))
144	2, 143	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺))
145	144	feqmptd 6960	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (invg‘𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg‘𝐺)‘𝑥)))
146	1, 5, 142	symginv 19269	. . . . . . . . 9 ⊢ (𝑥 ∈ (Base‘𝐺) → ((invg‘𝐺)‘𝑥) = ◡𝑥)
147	146	adantl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑥) = ◡𝑥)
148	32	feqmptd 6960	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → ◡𝑥 = (𝑦 ∈ 𝐴 ↦ (◡𝑥‘𝑦)))
149	147, 148	eqtrd 2772	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑥) = (𝑦 ∈ 𝐴 ↦ (◡𝑥‘𝑦)))
150	149	mpteq2dva 5248	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝐺) ↦ ((invg‘𝐺)‘𝑥)) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑦 ∈ 𝐴 ↦ (◡𝑥‘𝑦))))
151	145, 150	eqtrd 2772	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (invg‘𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑦 ∈ 𝐴 ↦ (◡𝑥‘𝑦))))
152		xkopt 23158	. . . . . . 7 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝒫 𝐴 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
153	26, 152	mpancom 686	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
154	153	oveq2d 7424	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((TopOpen‘𝐺) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)) = ((TopOpen‘𝐺) Cn (∏t‘(𝐴 × {𝒫 𝐴}))))
155	141, 151, 154	3eltr4d 2848	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)))
156		eqid 2732	. . . . . . 7 ⊢ (𝒫 𝐴 ↑ko 𝒫 𝐴) = (𝒫 𝐴 ↑ko 𝒫 𝐴)
157	156	xkotopon 23103	. . . . . 6 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ Top) → (𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
158	26, 26, 157	syl2anc 584	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
159		frn 6724	. . . . . 6 ⊢ ((invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺) → ran (invg‘𝐺) ⊆ (Base‘𝐺))
160	2, 143, 159	3syl 18	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ran (invg‘𝐺) ⊆ (Base‘𝐺))
161		cndis 22794	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴 ↑m 𝐴))
162	12, 161	mpdan 685	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴 ↑m 𝐴))
163	21, 162	sseqtrrd 4023	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝐺) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
164		cnrest2 22789	. . . . 5 ⊢ (((𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)) ∧ ran (invg‘𝐺) ⊆ (Base‘𝐺) ∧ (Base‘𝐺) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴)) → ((invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)) ↔ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝐺)))))
165	158, 160, 163, 164	syl3anc 1371	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)) ↔ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝐺)))))
166	155, 165	mpbid 231	. . 3 ⊢ (𝐴 ∈ 𝑉 → (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝐺))))
167	153	oveq1d 7423	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝐺)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)))
168	167, 11	eqtrd 2772	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝐺)) = (TopOpen‘𝐺))
169	168	oveq2d 7424	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((TopOpen‘𝐺) Cn ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝐺))) = ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
170	166, 169	eleqtrd 2835	. 2 ⊢ (𝐴 ∈ 𝑉 → (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
171		eqid 2732	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
172	171, 142	istgp 23580	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
173	2, 9, 170, 172	syl3anbrc 1343	1 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ TopGrp)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  {crab 3432  Vcvv 3474   ⊆ wss 3948  𝒫 cpw 4602  {csn 4628  ∪ cuni 4908   ↦ cmpt 5231   × cxp 5674  ^◡ccnv 5675  dom cdm 5676  ran crn 5677   “ cima 5679  Fun wfun 6537  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7408   ↑_m cmap 8819  Basecbs 17143   ↾_t crest 17365  TopOpenctopn 17366  ∏_tcpt 17383  SubMndcsubmnd 18669  EndoFMndcefmnd 18748  Grpcgrp 18818  inv_gcminusg 18819  SymGrpcsymg 19233  Topctop 22394  TopOnctopon 22411   Cn ccn 22727   CnP ccnp 22728   ↑_ko cxko 23064  TopMndctmd 23573  TopGrpctgp 23574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-ixp 8891  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-fi 9405  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-tset 17215  df-rest 17367  df-topn 17368  df-0g 17386  df-topgen 17388  df-pt 17389  df-plusf 18559  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-submnd 18671  df-efmnd 18749  df-grp 18821  df-minusg 18822  df-symg 19234  df-top 22395  df-topon 22412  df-topsp 22434  df-bases 22448  df-ntr 22523  df-nei 22601  df-cn 22730  df-cnp 22731  df-cmp 22890  df-lly 22969  df-nlly 22970  df-tx 23065  df-xko 23066  df-tmd 23575  df-tgp 23576

This theorem is referenced by: (None)