Theorem symgvalstruct 19366

Description: The value of the symmetric group function at 𝐴 represented as extensible structure with three slots. This corresponds to the former definition of SymGrp. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 31-Mar-2024.) (Proof shortened by AV, 6-Nov-2024.)

Hypotheses

Ref	Expression
symgvalstruct.g	⊢ 𝐺 = (SymGrp‘𝐴)
symgvalstruct.b	⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
symgvalstruct.m	⊢ 𝑀 = (𝐴 ↑m 𝐴)
symgvalstruct.p	⊢ + = (𝑓 ∈ 𝑀, 𝑔 ∈ 𝑀 ↦ (𝑓 ∘ 𝑔))
symgvalstruct.j	⊢ 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))

Assertion

Ref	Expression
symgvalstruct	⊢ (𝐴 ∈ 𝑉 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})

Distinct variable groups:   𝐴,𝑓,𝑔   𝑥,𝐴   𝑥,𝐵   𝑥,𝐺   𝑥,𝐽   𝑓,𝑀,𝑔   𝑥,𝑉   𝑥, +

Allowed substitution hints:   𝐵(𝑓,𝑔)   + (𝑓,𝑔)   𝐺(𝑓,𝑔)   𝐽(𝑓,𝑔)   𝑀(𝑥)   𝑉(𝑓,𝑔)

Proof of Theorem symgvalstruct

Step	Hyp	Ref	Expression
1		hashv01gt1 14302	. 2 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
2		hasheq0 14320	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
3		0symgefmndeq 19363	. . . . . . . . 9 ⊢ (EndoFMnd‘∅) = (SymGrp‘∅)
4	3	eqcomi 2750	. . . . . . . 8 ⊢ (SymGrp‘∅) = (EndoFMnd‘∅)
5		symgvalstruct.g	. . . . . . . . 9 ⊢ 𝐺 = (SymGrp‘𝐴)
6		fveq2 6830	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (SymGrp‘𝐴) = (SymGrp‘∅))
7	5, 6	eqtrid 2788	. . . . . . . 8 ⊢ (𝐴 = ∅ → 𝐺 = (SymGrp‘∅))
8		fveq2 6830	. . . . . . . 8 ⊢ (𝐴 = ∅ → (EndoFMnd‘𝐴) = (EndoFMnd‘∅))
9	4, 7, 8	3eqtr4a 2802	. . . . . . 7 ⊢ (𝐴 = ∅ → 𝐺 = (EndoFMnd‘𝐴))
10	9	adantl 483	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → 𝐺 = (EndoFMnd‘𝐴))
11		eqid 2741	. . . . . . . 8 ⊢ (EndoFMnd‘𝐴) = (EndoFMnd‘𝐴)
12		symgvalstruct.m	. . . . . . . 8 ⊢ 𝑀 = (𝐴 ↑m 𝐴)
13		symgvalstruct.p	. . . . . . . 8 ⊢ + = (𝑓 ∈ 𝑀, 𝑔 ∈ 𝑀 ↦ (𝑓 ∘ 𝑔))
14		symgvalstruct.j	. . . . . . . 8 ⊢ 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
15	11, 12, 13, 14	efmnd 18833	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
16	15	adantr 482	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
17		0map0sn0 8827	. . . . . . . . . . 11 ⊢ (∅ ↑m ∅) = {∅}
18		id 22	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
19	18, 18	oveq12d 7377	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → (𝐴 ↑m 𝐴) = (∅ ↑m ∅))
20		symgvalstruct.b	. . . . . . . . . . . 12 ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
21	7	fveq2d 6834	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → (Base‘𝐺) = (Base‘(SymGrp‘∅)))
22		eqid 2741	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐺) = (Base‘𝐺)
23	5, 22	symgbas 19341	. . . . . . . . . . . . 13 ⊢ (Base‘𝐺) = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
24		symgbas0 19358	. . . . . . . . . . . . 13 ⊢ (Base‘(SymGrp‘∅)) = {∅}
25	21, 23, 24	3eqtr3g 2799	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} = {∅})
26	20, 25	eqtrid 2788	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → 𝐵 = {∅})
27	17, 19, 26	3eqtr4a 2802	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ↑m 𝐴) = 𝐵)
28	27	adantl 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → (𝐴 ↑m 𝐴) = 𝐵)
29	12, 28	eqtrid 2788	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → 𝑀 = 𝐵)
30	29	opeq2d 4813	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
31	30	tpeq1d 4679	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
32	10, 16, 31	3eqtrd 2780	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
33	32	ex 414	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ∅ → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
34	2, 33	sylbid 242	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
35		hash1snb 14376	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 ↔ ∃𝑥 𝐴 = {𝑥}))
36		vsnex 5366	. . . . . . . 8 ⊢ {𝑥} ∈ V
37		eleq1 2829	. . . . . . . 8 ⊢ (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
38	36, 37	mpbiri 260	. . . . . . 7 ⊢ (𝐴 = {𝑥} → 𝐴 ∈ V)
39	11, 12, 13, 14	efmnd 18833	. . . . . . 7 ⊢ (𝐴 ∈ V → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
40	38, 39	syl 17	. . . . . 6 ⊢ (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
41		snsymgefmndeq 19364	. . . . . . 7 ⊢ (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))
42	41, 5	eqtr4di 2794	. . . . . 6 ⊢ (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = 𝐺)
43	42	fveq2d 6834	. . . . . . . . 9 ⊢ (𝐴 = {𝑥} → (Base‘(EndoFMnd‘𝐴)) = (Base‘𝐺))
44		eqid 2741	. . . . . . . . . . 11 ⊢ (Base‘(EndoFMnd‘𝐴)) = (Base‘(EndoFMnd‘𝐴))
45	11, 44	efmndbas 18834	. . . . . . . . . 10 ⊢ (Base‘(EndoFMnd‘𝐴)) = (𝐴 ↑m 𝐴)
46	45, 12	eqtr4i 2767	. . . . . . . . 9 ⊢ (Base‘(EndoFMnd‘𝐴)) = 𝑀
47	23, 20	eqtr4i 2767	. . . . . . . . 9 ⊢ (Base‘𝐺) = 𝐵
48	43, 46, 47	3eqtr3g 2799	. . . . . . . 8 ⊢ (𝐴 = {𝑥} → 𝑀 = 𝐵)
49	48	opeq2d 4813	. . . . . . 7 ⊢ (𝐴 = {𝑥} → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
50	49	tpeq1d 4679	. . . . . 6 ⊢ (𝐴 = {𝑥} → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
51	40, 42, 50	3eqtr3d 2784	. . . . 5 ⊢ (𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
52	51	exlimiv 1938	. . . 4 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
53	35, 52	biimtrdi 255	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
54		ssnpss 4039	. . . . . . 7 ⊢ ((𝐴 ↑m 𝐴) ⊆ 𝐵 → ¬ 𝐵 ⊊ (𝐴 ↑m 𝐴))
55	11, 5	symgpssefmnd 19365	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
56	20, 23	eqtr4i 2767	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
57	45	eqcomi 2750	. . . . . . . . 9 ⊢ (𝐴 ↑m 𝐴) = (Base‘(EndoFMnd‘𝐴))
58	56, 57	psseq12i 4027	. . . . . . . 8 ⊢ (𝐵 ⊊ (𝐴 ↑m 𝐴) ↔ (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
59	55, 58	sylibr 236	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ⊊ (𝐴 ↑m 𝐴))
60	54, 59	nsyl3 138	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ¬ (𝐴 ↑m 𝐴) ⊆ 𝐵)
61		fvexd 6845	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) ∈ V)
62		f1osetex 8800	. . . . . . . 8 ⊢ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ∈ V
63	20, 62	eqeltri 2837	. . . . . . 7 ⊢ 𝐵 ∈ V
64	63	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ∈ V)
65	5, 20	symgval 19340	. . . . . . 7 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)
66	65, 57	ressval2 17200	. . . . . 6 ⊢ ((¬ (𝐴 ↑m 𝐴) ⊆ 𝐵 ∧ (EndoFMnd‘𝐴) ∈ V ∧ 𝐵 ∈ V) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩))
67	60, 61, 64, 66	syl3anc 1380	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩))
68		ovex 7392	. . . . . . 7 ⊢ (𝐴 ↑m 𝐴) ∈ V
69	68	inex2 5248	. . . . . 6 ⊢ (𝐵 ∩ (𝐴 ↑m 𝐴)) ∈ V
70		setsval 17132	. . . . . 6 ⊢ (((EndoFMnd‘𝐴) ∈ V ∧ (𝐵 ∩ (𝐴 ↑m 𝐴)) ∈ V) → ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩) = (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}))
71	61, 69, 70	sylancl 593	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩) = (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}))
72	15	adantr 482	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
73	72	reseq1d 5936	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) = ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})))
74	73	uneq1d 4099	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}) = (({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}))
75		eqidd 2742	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
76		fvexd 6845	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ∈ V)
77		fvexd 6845	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ∈ V)
78	12, 68	eqeltri 2837	. . . . . . . . . . 11 ⊢ 𝑀 ∈ V
79	78, 78	mpoex 8023	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑀, 𝑔 ∈ 𝑀 ↦ (𝑓 ∘ 𝑔)) ∈ V
80	13, 79	eqeltri 2837	. . . . . . . . 9 ⊢ + ∈ V
81	80	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → + ∈ V)
82	14	fvexi 6844	. . . . . . . . 9 ⊢ 𝐽 ∈ V
83	82	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐽 ∈ V)
84		basendxnplusgndx 17245	. . . . . . . . . 10 ⊢ (Base‘ndx) ≠ (+g‘ndx)
85	84	necomi 2990	. . . . . . . . 9 ⊢ (+g‘ndx) ≠ (Base‘ndx)
86	85	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ≠ (Base‘ndx))
87		tsetndxnbasendx 17314	. . . . . . . . 9 ⊢ (TopSet‘ndx) ≠ (Base‘ndx)
88	87	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ≠ (Base‘ndx))
89	75, 76, 77, 81, 83, 86, 88	tpres 7148	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) = {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
90	89	uneq1d 4099	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}) = ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}))
91		uncom 4090	. . . . . . . 8 ⊢ ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}) = ({⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩} ∪ {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
92		tpass 4686	. . . . . . . 8 ⊢ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = ({⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩} ∪ {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
93	91, 92	eqtr4i 2767	. . . . . . 7 ⊢ ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}) = {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}
94	5, 56	symgbasmap 19346	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐴 ↑m 𝐴))
95	94	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐴 ↑m 𝐴)))
96	95	ssrdv 3922	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ⊆ (𝐴 ↑m 𝐴))
97		dfss2 3902	. . . . . . . . . 10 ⊢ (𝐵 ⊆ (𝐴 ↑m 𝐴) ↔ (𝐵 ∩ (𝐴 ↑m 𝐴)) = 𝐵)
98	96, 97	sylib 220	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (𝐵 ∩ (𝐴 ↑m 𝐴)) = 𝐵)
99	98	opeq2d 4813	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩ = ⟨(Base‘ndx), 𝐵⟩)
100	99	tpeq1d 4679	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
101	93, 100	eqtrid 2788	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
102	74, 90, 101	3eqtrd 2780	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
103	67, 71, 102	3eqtrd 2780	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
104	103	ex 414	. . 3 ⊢ (𝐴 ∈ 𝑉 → (1 < (♯‘𝐴) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
105	34, 53, 104	3jaod 1438	. 2 ⊢ (𝐴 ∈ 𝑉 → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
106	1, 105	mpd 15	1 ⊢ (𝐴 ∈ 𝑉 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∨ w3o 1092   = wceq 1548  ∃wex 1787   ∈ wcel 2121  {cab 2719   ≠ wne 2936  Vcvv 3433   ∖ cdif 3881   ∪ cun 3882   ∩ cin 3883   ⊆ wss 3884   ⊊ wpss 3885  ∅c0 4263  𝒫 cpw 4531  {csn 4557  {cpr 4559  {ctp 4561  ⟨cop 4563   class class class wbr 5074   × cxp 5618   ↾ cres 5622   ∘ ccom 5624  –1-1-onto→wf1o 6487  ‘cfv 6488  (class class class)co 7359   ∈ cmpo 7361   ↑_m cmap 8767  0cc0 11034  1c1 11035   < clt 11175  ♯chash 14287   sSet csts 17128  ndxcnx 17158  Basecbs 17174  +_gcplusg 17215  TopSetcts 17221  ∏_tcpt 17396  EndoFMndcefmnd 18831  SymGrpcsymg 19338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681  ax-cnex 11090  ax-resscn 11091  ax-1cn 11092  ax-icn 11093  ax-addcl 11094  ax-addrcl 11095  ax-mulcl 11096  ax-mulrcl 11097  ax-mulcom 11098  ax-addass 11099  ax-mulass 11100  ax-distr 11101  ax-i2m1 11102  ax-1ne0 11103  ax-1rid 11104  ax-rnegex 11105  ax-rrecex 11106  ax-cnre 11107  ax-pre-lttri 11108  ax-pre-lttrn 11109  ax-pre-ltadd 11110  ax-pre-mulgt0 11111

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-om 7810  df-1st 7933  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-1o 8399  df-oadd 8403  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9820  df-card 9858  df-pnf 11177  df-mnf 11178  df-xr 11179  df-ltxr 11180  df-le 11181  df-sub 11375  df-neg 11376  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-xnn0 12506  df-z 12520  df-uz 12784  df-fz 13457  df-hash 14288  df-struct 17112  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-tset 17234  df-efmnd 18832  df-symg 19339

This theorem is referenced by: (None)