Theorem symgtgp 24333

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 19528	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ Grp)
3		eqid 2760	. . . 4 ⊢ (EndoFMnd‘𝐴) = (EndoFMnd‘𝐴)
4	3	efmndtmd 24328	. . 3 ⊢ (𝐴 ∈ 𝑉 → (EndoFMnd‘𝐴) ∈ TopMnd)
5		eqid 2760	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	3, 1, 5	symgsubmefmnd 19526	. . 3 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝐺) ∈ (SubMnd‘(EndoFMnd‘𝐴)))
7	1, 5, 3	symgressbas 19510	. . . 4 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s (Base‘𝐺))
8	7	submtmd 24331	. . 3 ⊢ (((EndoFMnd‘𝐴) ∈ TopMnd ∧ (Base‘𝐺) ∈ (SubMnd‘(EndoFMnd‘𝐴))) → 𝐺 ∈ TopMnd)
9	4, 6, 8	syl2anc 596	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ TopMnd)
10		eqid 2760	. . . . . 6 ⊢ (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
11	1, 5	symgtopn 19534	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) = (TopOpen‘𝐺))
12		distopon 23223	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴))
13	10	pttoponconst 23824	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)))
14	12, 13	mpdan 700	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)))
15	1, 5	elsymgbas 19502	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝐺) ↔ 𝑥:𝐴–1-1-onto→𝐴))
16		f1of 6818	. . . . . . . . . . 11 ⊢ (𝑥:𝐴–1-1-onto→𝐴 → 𝑥:𝐴⟶𝐴)
17		elmapg 8839	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ (𝐴 ↑m 𝐴) ↔ 𝑥:𝐴⟶𝐴))
18	17	anidms 577	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (𝐴 ↑m 𝐴) ↔ 𝑥:𝐴⟶𝐴))
19	16, 18	imbitrrid 249	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → (𝑥:𝐴–1-1-onto→𝐴 → 𝑥 ∈ (𝐴 ↑m 𝐴)))
20	15, 19	sylbid 243	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝐺) → 𝑥 ∈ (𝐴 ↑m 𝐴)))
21	20	ssrdv 3937	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝐺) ⊆ (𝐴 ↑m 𝐴))
22		resttopon 23387	. . . . . . . 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 596	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) ∈ (TopOn‘(Base‘𝐺)))
24	11, 23	eqeltrrd 2861	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
25		id 23	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
26		distop 23221	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
27		fconst6g 6765	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ Top → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
28	26, 27	syl 18	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
29	15	biimpa 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥:𝐴–1-1-onto→𝐴)
30		f1ocnv 6831	. . . . . . . . . . . 12 ⊢ (𝑥:𝐴–1-1-onto→𝐴 → ◡𝑥:𝐴–1-1-onto→𝐴)
31		f1of 6818	. . . . . . . . . . . 12 ⊢ (◡𝑥:𝐴–1-1-onto→𝐴 → ◡𝑥:𝐴⟶𝐴)
32	29, 30, 31	3syl 19	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → ◡𝑥:𝐴⟶𝐴)
33	32	ffvelcdmda 7078	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ 𝐴) → (◡𝑥‘𝑦) ∈ 𝐴)
34	33	an32s 665	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ (Base‘𝐺)) → (◡𝑥‘𝑦) ∈ 𝐴)
35	34	fmpttd 7109	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)):(Base‘𝐺)⟶𝐴)
36	35	adantr 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)):(Base‘𝐺)⟶𝐴)
37		cnveq 5853	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑓 → ◡𝑥 = ◡𝑓)
38	37	fveq1d 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑓 → (◡𝑥‘𝑦) = (◡𝑓‘𝑦))
39		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) = (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦))
40		fvex 6892	. . . . . . . . . . . . . . 15 ⊢ (◡𝑓‘𝑦) ∈ V
41	38, 39, 40	fvmpt 6987	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (Base‘𝐺) → ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦))‘𝑓) = (◡𝑓‘𝑦))
42	41	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → ((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦))‘𝑓) = (◡𝑓‘𝑦))
43	42	eleq1d 2845	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ 𝒫 𝐴) → (((𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦))‘𝑓) ∈ 𝑡 ↔ (◡𝑓‘𝑦) ∈ 𝑡))
44		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(◡𝑓‘𝑦))) = (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(◡𝑓‘𝑦)))
45	44	mptiniseg 6235	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ V → (◡(𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(◡𝑓‘𝑦))) “ {𝑦}) = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦})
46	45	elv 3455	. . . . . . . . . . . . . . . 16 ⊢ (◡(𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(◡𝑓‘𝑦))) “ {𝑦}) = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦}
47		eqid 2760	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺))
48	14	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)))
49	21	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (Base‘𝐺) ⊆ (𝐴 ↑m 𝐴))
50		toponuni 23140	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∏t‘(𝐴 × {𝒫 𝐴})) ∈ (TopOn‘(𝐴 ↑m 𝐴)) → (𝐴 ↑m 𝐴) = ∪ (∏t‘(𝐴 × {𝒫 𝐴})))
51		mpteq1 5194	. . . . . . . . . . . . . . . . . . . . 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 19	. . . . . . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝐴 ∈ 𝑉)
54	28	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝐴 × {𝒫 𝐴}):𝐴⟶Top)
55	1, 5	elsymgbas 19502	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (Base‘𝐺) ↔ 𝑓:𝐴–1-1-onto→𝐴))
56	55	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → (𝑓 ∈ (Base‘𝐺) ↔ 𝑓:𝐴–1-1-onto→𝐴))
57	56	biimpa 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓:𝐴–1-1-onto→𝐴)
58		f1ocnv 6831	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓:𝐴–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→𝐴)
59		f1of 6818	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (◡𝑓:𝐴–1-1-onto→𝐴 → ◡𝑓:𝐴⟶𝐴)
60	57, 58, 59	3syl 19	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ◡𝑓:𝐴⟶𝐴)
61		simplr 781	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑦 ∈ 𝐴)
62	60, 61	ffvelcdmd 7079	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (◡𝑓‘𝑦) ∈ 𝐴)
63		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∪ (∏t‘(𝐴 × {𝒫 𝐴})) = ∪ (∏t‘(𝐴 × {𝒫 𝐴}))
64	63, 10	ptpjcn 23838	. . . . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → ((𝐴 × {𝒫 𝐴})‘(◡𝑓‘𝑦)) = 𝒫 𝐴)
69	68	oveq2d 7430	. . . . . . . . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . . . . . . . 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 2860	. . . . . . . . . . . . . . . . . . 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 23903	. . . . . . . . . . . . . . . . . 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 7429	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ 𝑉 → (((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)) Cn 𝒫 𝐴) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
74	73	ad2antrr 739	. . . . . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(◡𝑓‘𝑦))) ∈ ((TopOpen‘𝐺) Cn 𝒫 𝐴))
76		snelpwi 5419	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝒫 𝐴)
77	76	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → {𝑦} ∈ 𝒫 𝐴)
78		cnima 23491	. . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (◡(𝑢 ∈ (Base‘𝐺) ↦ (𝑢‘(◡𝑓‘𝑦))) “ {𝑦}) ∈ (TopOpen‘𝐺))
80	46, 79	eqeltrrid 2865	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} ∈ (TopOpen‘𝐺))
81	80	adantr 486	. . . . . . . . . . . . . 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 6878	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑓 → (𝑢‘(◡𝑓‘𝑦)) = (𝑓‘(◡𝑓‘𝑦)))
83	82	eqeq1d 2762	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑓 → ((𝑢‘(◡𝑓‘𝑦)) = 𝑦 ↔ (𝑓‘(◡𝑓‘𝑦)) = 𝑦))
84		simplr 781	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → 𝑓 ∈ (Base‘𝐺))
85	57	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → 𝑓:𝐴–1-1-onto→𝐴)
86		simpllr 788	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → 𝑦 ∈ 𝐴)
87		f1ocnvfv2 7279	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:𝐴–1-1-onto→𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑓‘(◡𝑓‘𝑦)) = 𝑦)
88	85, 86, 87	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → (𝑓‘(◡𝑓‘𝑦)) = 𝑦)
89	83, 84, 88	elrabd 3647	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) → 𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦})
90		ssrab2 4028	. . . . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → (◡𝑓‘𝑦) ∈ 𝐴)
95		f1ocnvfv 7280	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥:𝐴–1-1-onto→𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝐴) → ((𝑥‘(◡𝑓‘𝑦)) = 𝑦 → (◡𝑥‘𝑦) = (◡𝑓‘𝑦)))
96	93, 94, 95	syl2anc 596	. . . . . . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) ∧ (𝑡 ∈ 𝒫 𝐴 ∧ (◡𝑓‘𝑦) ∈ 𝑡)) ∧ 𝑥 ∈ (Base‘𝐺)) → (◡𝑓‘𝑦) ∈ 𝑡)
98		eleq1 2848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((◡𝑥‘𝑦) = (◡𝑓‘𝑦) → ((◡𝑥‘𝑦) ∈ 𝑡 ↔ (◡𝑓‘𝑦) ∈ 𝑡))
99	97, 98	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . 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 48	. . . . . . . . . . . . . . . . . . 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 3154	. . . . . . . . . . . . . . . . . 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 6878	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑥 → (𝑢‘(◡𝑓‘𝑦)) = (𝑥‘(◡𝑓‘𝑦)))
103	102	eqeq1d 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = 𝑥 → ((𝑢‘(◡𝑓‘𝑦)) = 𝑦 ↔ (𝑥‘(◡𝑓‘𝑦)) = 𝑦))
104	103	ralrab 3652	. . . . . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . . . . 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 4019	. . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . 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 6234	. . . . . . . . . . . . . . . 16 ⊢ (◡(𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) “ 𝑡) = {𝑥 ∈ (Base‘𝐺) ∣ (◡𝑥‘𝑦) ∈ 𝑡}
109	107, 108	sseqtrrdi 3972	. . . . . . . . . . . . . . 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 6572	. . . . . . . . . . . . . . . 16 ⊢ Fun (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦))
111		fvex 6892	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑥‘𝑦) ∈ V
112	111, 39	dmmpti 6677	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) = (Base‘𝐺)
113	91, 112	sseqtrrdi 3972	. . . . . . . . . . . . . . . 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 7047	. . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . 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 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦} → (𝑓 ∈ 𝑣 ↔ 𝑓 ∈ {𝑢 ∈ (Base‘𝐺) ∣ (𝑢‘(◡𝑓‘𝑦)) = 𝑦}))
118		imaeq2 6052	. . . . . . . . . . . . . . . . 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 3962	. . . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . . 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 3576	. . . . . . . . . . . . . 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 850	. . . . . . . . . . . . 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 462	. . . . . . . . . . . 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 243	. . . . . . . . . . 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 3154	. . . . . . . . . 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 739	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
127	12	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝒫 𝐴 ∈ (TopOn‘𝐴))
128		simpr 490	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → 𝑓 ∈ (Base‘𝐺))
129		iscnp 23463	. . . . . . . . . . 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 1398	. . . . . . . . . 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 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑓 ∈ (Base‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))
132	131	ralrimiva 3154	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ∀𝑓 ∈ (Base‘𝐺)(𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∈ (((TopOpen‘𝐺) CnP 𝒫 𝐴)‘𝑓))
133		cncnp 23506	. . . . . . . . . 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 596	. . . . . . . . 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 486	. . . . . . . 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 726	. . . . . . 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 592	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ((𝐴 × {𝒫 𝐴})‘𝑦) = 𝒫 𝐴)
139	138	oveq2d 7430	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ((TopOpen‘𝐺) Cn ((𝐴 × {𝒫 𝐴})‘𝑦)) = ((TopOpen‘𝐺) Cn 𝒫 𝐴))
140	136, 139	eleqtrrd 2863	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ (Base‘𝐺) ↦ (◡𝑥‘𝑦)) ∈ ((TopOpen‘𝐺) Cn ((𝐴 × {𝒫 𝐴})‘𝑦)))
141	10, 24, 25, 28, 140	ptcn 23854	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝐺) ↦ (𝑦 ∈ 𝐴 ↦ (◡𝑥‘𝑦))) ∈ ((TopOpen‘𝐺) Cn (∏t‘(𝐴 × {𝒫 𝐴}))))
142		eqid 2760	. . . . . . . . 9 ⊢ (invg‘𝐺) = (invg‘𝐺)
143	5, 142	grpinvf 19111	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺))
144	2, 143	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺))
145	144	feqmptd 6947	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (invg‘𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg‘𝐺)‘𝑥)))
146	1, 5, 142	symginv 19530	. . . . . . . . 9 ⊢ (𝑥 ∈ (Base‘𝐺) → ((invg‘𝐺)‘𝑥) = ◡𝑥)
147	146	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑥) = ◡𝑥)
148	32	feqmptd 6947	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → ◡𝑥 = (𝑦 ∈ 𝐴 ↦ (◡𝑥‘𝑦)))
149	147, 148	eqtrd 2795	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝐺)) → ((invg‘𝐺)‘𝑥) = (𝑦 ∈ 𝐴 ↦ (◡𝑥‘𝑦)))
150	149	mpteq2dva 5198	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (Base‘𝐺) ↦ ((invg‘𝐺)‘𝑥)) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑦 ∈ 𝐴 ↦ (◡𝑥‘𝑦))))
151	145, 150	eqtrd 2795	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (invg‘𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ (𝑦 ∈ 𝐴 ↦ (◡𝑥‘𝑦))))
152		xkopt 23882	. . . . . . 7 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝐴 ∈ 𝑉) → (𝒫 𝐴 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
153	26, 152	mpancom 701	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ↑ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝒫 𝐴})))
154	153	oveq2d 7430	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((TopOpen‘𝐺) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)) = ((TopOpen‘𝐺) Cn (∏t‘(𝐴 × {𝒫 𝐴}))))
155	141, 151, 154	3eltr4d 2875	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (𝒫 𝐴 ↑ko 𝒫 𝐴)))
156		eqid 2760	. . . . . . 7 ⊢ (𝒫 𝐴 ↑ko 𝒫 𝐴) = (𝒫 𝐴 ↑ko 𝒫 𝐴)
157	156	xkotopon 23827	. . . . . 6 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐴 ∈ Top) → (𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
158	26, 26, 157	syl2anc 596	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ↑ko 𝒫 𝐴) ∈ (TopOn‘(𝒫 𝐴 Cn 𝒫 𝐴)))
159		frn 6711	. . . . . 6 ⊢ ((invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺) → ran (invg‘𝐺) ⊆ (Base‘𝐺))
160	2, 143, 159	3syl 19	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ran (invg‘𝐺) ⊆ (Base‘𝐺))
161		cndis 23517	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝒫 𝐴 ∈ (TopOn‘𝐴)) → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴 ↑m 𝐴))
162	12, 161	mpdan 700	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 Cn 𝒫 𝐴) = (𝐴 ↑m 𝐴))
163	21, 162	sseqtrrd 3968	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (Base‘𝐺) ⊆ (𝒫 𝐴 Cn 𝒫 𝐴))
164		cnrest2 23512	. . . . 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 1398	. . . 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 235	. . 3 ⊢ (𝐴 ∈ 𝑉 → (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝐺))))
167	153	oveq1d 7429	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝐺)) = ((∏t‘(𝐴 × {𝒫 𝐴})) ↾t (Base‘𝐺)))
168	167, 11	eqtrd 2795	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝐺)) = (TopOpen‘𝐺))
169	168	oveq2d 7430	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((TopOpen‘𝐺) Cn ((𝒫 𝐴 ↑ko 𝒫 𝐴) ↾t (Base‘𝐺))) = ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
170	166, 169	eleqtrd 2862	. 2 ⊢ (𝐴 ∈ 𝑉 → (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
171		eqid 2760	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
172	171, 142	istgp 24304	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
173	2, 9, 170, 172	syl3anbrc 1362	1 ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ TopGrp)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3899  𝒫 cpw 4557  {csn 4584  ∪ cuni 4867   ↦ cmpt 5186   × cxp 5653  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658  Fun wfun 6527  ⟶wf 6529  –1-1-onto→wf1o 6532  ‘cfv 6533  (class class class)co 7414   ↑_m cmap 8827  Basecbs 17302   ↾_t crest 17506  TopOpenctopn 17507  ∏_tcpt 17524  SubMndcsubmnd 18891  EndoFMndcefmnd 18978  Grpcgrp 19058  inv_gcminusg 19059  SymGrpcsymg 19497  Topctop 23119  TopOnctopon 23136   Cn ccn 23450   CnP ccnp 23451   ↑_ko cxko 23788  TopMndctmd 24297  TopGrpctgp 24298

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-2o 8457  df-er 8697  df-map 8829  df-ixp 8906  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-fi 9382  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-nn 12259  df-2 12328  df-3 12329  df-4 12330  df-5 12331  df-6 12332  df-7 12333  df-8 12334  df-9 12335  df-n0 12530  df-z 12617  df-uz 12889  df-fz 13563  df-struct 17240  df-sets 17257  df-slot 17275  df-ndx 17287  df-base 17303  df-ress 17324  df-plusg 17356  df-tset 17362  df-rest 17508  df-topn 17509  df-0g 17527  df-topgen 17529  df-pt 17530  df-plusf 18730  df-mgm 18731  df-sgrp 18822  df-mnd 18838  df-submnd 18893  df-efmnd 18979  df-grp 19061  df-minusg 19062  df-symg 19498  df-top 23120  df-topon 23137  df-topsp 23159  df-bases 23172  df-ntr 23246  df-nei 23324  df-cn 23453  df-cnp 23454  df-cmp 23613  df-lly 23693  df-nlly 23694  df-tx 23789  df-xko 23790  df-tmd 24299  df-tgp 24300

This theorem is used by: (None)