Theorem symgvalstruct 19004

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 14059	. 2 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
2		hasheq0 14078	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
3		0symgefmndeq 19001	. . . . . . . . 9 ⊢ (EndoFMnd‘∅) = (SymGrp‘∅)
4	3	eqcomi 2747	. . . . . . . 8 ⊢ (SymGrp‘∅) = (EndoFMnd‘∅)
5		symgvalstruct.g	. . . . . . . . 9 ⊢ 𝐺 = (SymGrp‘𝐴)
6		fveq2 6774	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (SymGrp‘𝐴) = (SymGrp‘∅))
7	5, 6	eqtrid 2790	. . . . . . . 8 ⊢ (𝐴 = ∅ → 𝐺 = (SymGrp‘∅))
8		fveq2 6774	. . . . . . . 8 ⊢ (𝐴 = ∅ → (EndoFMnd‘𝐴) = (EndoFMnd‘∅))
9	4, 7, 8	3eqtr4a 2804	. . . . . . 7 ⊢ (𝐴 = ∅ → 𝐺 = (EndoFMnd‘𝐴))
10	9	adantl 482	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → 𝐺 = (EndoFMnd‘𝐴))
11		eqid 2738	. . . . . . . 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 18509	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
16	15	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
17		0map0sn0 8673	. . . . . . . . . . 11 ⊢ (∅ ↑m ∅) = {∅}
18		id 22	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
19	18, 18	oveq12d 7293	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → (𝐴 ↑m 𝐴) = (∅ ↑m ∅))
20		symgvalstruct.b	. . . . . . . . . . . 12 ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
21	7	fveq2d 6778	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → (Base‘𝐺) = (Base‘(SymGrp‘∅)))
22		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐺) = (Base‘𝐺)
23	5, 22	symgbas 18978	. . . . . . . . . . . . 13 ⊢ (Base‘𝐺) = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
24		symgbas0 18996	. . . . . . . . . . . . 13 ⊢ (Base‘(SymGrp‘∅)) = {∅}
25	21, 23, 24	3eqtr3g 2801	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} = {∅})
26	20, 25	eqtrid 2790	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → 𝐵 = {∅})
27	17, 19, 26	3eqtr4a 2804	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ↑m 𝐴) = 𝐵)
28	27	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → (𝐴 ↑m 𝐴) = 𝐵)
29	12, 28	eqtrid 2790	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → 𝑀 = 𝐵)
30	29	opeq2d 4811	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
31	30	tpeq1d 4681	. . . . . 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 2782	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
33	32	ex 413	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ∅ → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
34	2, 33	sylbid 239	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
35		hash1snb 14134	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 ↔ ∃𝑥 𝐴 = {𝑥}))
36		snex 5354	. . . . . . . 8 ⊢ {𝑥} ∈ V
37		eleq1 2826	. . . . . . . 8 ⊢ (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
38	36, 37	mpbiri 257	. . . . . . 7 ⊢ (𝐴 = {𝑥} → 𝐴 ∈ V)
39	11, 12, 13, 14	efmnd 18509	. . . . . . 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 19002	. . . . . . 7 ⊢ (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))
42	41, 5	eqtr4di 2796	. . . . . 6 ⊢ (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = 𝐺)
43	42	fveq2d 6778	. . . . . . . . 9 ⊢ (𝐴 = {𝑥} → (Base‘(EndoFMnd‘𝐴)) = (Base‘𝐺))
44		eqid 2738	. . . . . . . . . . 11 ⊢ (Base‘(EndoFMnd‘𝐴)) = (Base‘(EndoFMnd‘𝐴))
45	11, 44	efmndbas 18510	. . . . . . . . . 10 ⊢ (Base‘(EndoFMnd‘𝐴)) = (𝐴 ↑m 𝐴)
46	45, 12	eqtr4i 2769	. . . . . . . . 9 ⊢ (Base‘(EndoFMnd‘𝐴)) = 𝑀
47	23, 20	eqtr4i 2769	. . . . . . . . 9 ⊢ (Base‘𝐺) = 𝐵
48	43, 46, 47	3eqtr3g 2801	. . . . . . . 8 ⊢ (𝐴 = {𝑥} → 𝑀 = 𝐵)
49	48	opeq2d 4811	. . . . . . 7 ⊢ (𝐴 = {𝑥} → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
50	49	tpeq1d 4681	. . . . . 6 ⊢ (𝐴 = {𝑥} → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
51	40, 42, 50	3eqtr3d 2786	. . . . 5 ⊢ (𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
52	51	exlimiv 1933	. . . 4 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
53	35, 52	syl6bi 252	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
54		ssnpss 4038	. . . . . . 7 ⊢ ((𝐴 ↑m 𝐴) ⊆ 𝐵 → ¬ 𝐵 ⊊ (𝐴 ↑m 𝐴))
55	11, 5	symgpssefmnd 19003	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
56	20, 23	eqtr4i 2769	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
57	45	eqcomi 2747	. . . . . . . . 9 ⊢ (𝐴 ↑m 𝐴) = (Base‘(EndoFMnd‘𝐴))
58	56, 57	psseq12i 4026	. . . . . . . 8 ⊢ (𝐵 ⊊ (𝐴 ↑m 𝐴) ↔ (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
59	55, 58	sylibr 233	. . . . . . 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 6789	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) ∈ V)
62		f1osetex 8647	. . . . . . . 8 ⊢ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ∈ V
63	20, 62	eqeltri 2835	. . . . . . 7 ⊢ 𝐵 ∈ V
64	63	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ∈ V)
65	5, 20	symgval 18976	. . . . . . 7 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)
66	65, 57	ressval2 16946	. . . . . 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 1370	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩))
68		ovex 7308	. . . . . . 7 ⊢ (𝐴 ↑m 𝐴) ∈ V
69	68	inex2 5242	. . . . . 6 ⊢ (𝐵 ∩ (𝐴 ↑m 𝐴)) ∈ V
70		setsval 16868	. . . . . 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 586	. . . . 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 481	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
73	72	reseq1d 5890	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) = ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})))
74	73	uneq1d 4096	. . . . . 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 2739	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
76		fvexd 6789	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ∈ V)
77		fvexd 6789	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ∈ V)
78	12, 68	eqeltri 2835	. . . . . . . . . . 11 ⊢ 𝑀 ∈ V
79	78, 78	mpoex 7920	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑀, 𝑔 ∈ 𝑀 ↦ (𝑓 ∘ 𝑔)) ∈ V
80	13, 79	eqeltri 2835	. . . . . . . . 9 ⊢ + ∈ V
81	80	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → + ∈ V)
82	14	fvexi 6788	. . . . . . . . 9 ⊢ 𝐽 ∈ V
83	82	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐽 ∈ V)
84		basendxnplusgndx 16992	. . . . . . . . . 10 ⊢ (Base‘ndx) ≠ (+g‘ndx)
85	84	necomi 2998	. . . . . . . . 9 ⊢ (+g‘ndx) ≠ (Base‘ndx)
86	85	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ≠ (Base‘ndx))
87		tsetndxnbasendx 17066	. . . . . . . . 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 7076	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) = {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
90	89	uneq1d 4096	. . . . . 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 4087	. . . . . . . 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 4688	. . . . . . . 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 2769	. . . . . . 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 18984	. . . . . . . . . . . 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 3927	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ⊆ (𝐴 ↑m 𝐴))
97		df-ss 3904	. . . . . . . . . 10 ⊢ (𝐵 ⊆ (𝐴 ↑m 𝐴) ↔ (𝐵 ∩ (𝐴 ↑m 𝐴)) = 𝐵)
98	96, 97	sylib 217	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (𝐵 ∩ (𝐴 ↑m 𝐴)) = 𝐵)
99	98	opeq2d 4811	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩ = ⟨(Base‘ndx), 𝐵⟩)
100	99	tpeq1d 4681	. . . . . . 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 2790	. . . . . 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 2782	. . . . 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 2782	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
104	103	ex 413	. . 3 ⊢ (𝐴 ∈ 𝑉 → (1 < (♯‘𝐴) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
105	34, 53, 104	3jaod 1427	. 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 396   ∨ w3o 1085   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715   ≠ wne 2943  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887   ⊊ wpss 3888  ∅c0 4256  𝒫 cpw 4533  {csn 4561  {cpr 4563  {ctp 4565  ⟨cop 4567   class class class wbr 5074   × cxp 5587   ↾ cres 5591   ∘ ccom 5593  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   ↑_m cmap 8615  0cc0 10871  1c1 10872   < clt 11009  ♯chash 14044   sSet csts 16864  ndxcnx 16894  Basecbs 16912  +_gcplusg 16962  TopSetcts 16968  ∏_tcpt 17149  EndoFMndcefmnd 18507  SymGrpcsymg 18974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-fz 13240  df-hash 14045  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-tset 16981  df-efmnd 18508  df-symg 18975

This theorem is referenced by: (None)