Theorem ushrisomgr 42577

Description: A simple hypergraph (with arbitrarily indexed edges) is isomorphic to a graph with the same vertices and the same edges, indexed by the edges themselves. (Contributed by AV, 11-Nov-2022.)

Hypotheses

Ref	Expression
ushrisomgr.v	⊢ 𝑉 = (Vtx‘𝐺)
ushrisomgr.e	⊢ 𝐸 = (Edg‘𝐺)
ushrisomgr.s	⊢ 𝐻 = ⟨𝑉, ( I ↾ 𝐸)⟩

Assertion

Ref	Expression
ushrisomgr	⊢ (𝐺 ∈ USHGraph → 𝐺 IsomGr 𝐻)

Proof of Theorem ushrisomgr

Dummy variables 𝑓 𝑔 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ushrisomgr.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	fvexi 6451	. . . . 5 ⊢ 𝑉 ∈ V
3	2	a1i 11	. . . 4 ⊢ (𝐺 ∈ USHGraph → 𝑉 ∈ V)
4	3	resiexd 6741	. . 3 ⊢ (𝐺 ∈ USHGraph → ( I ↾ 𝑉) ∈ V)
5		f1oi 6419	. . . . . 6 ⊢ ( I ↾ 𝑉):𝑉–1-1-onto→𝑉
6	5	a1i 11	. . . . 5 ⊢ (𝐺 ∈ USHGraph → ( I ↾ 𝑉):𝑉–1-1-onto→𝑉)
7		ushrisomgr.s	. . . . . . . 8 ⊢ 𝐻 = ⟨𝑉, ( I ↾ 𝐸)⟩
8	7	fveq2i 6440	. . . . . . 7 ⊢ (Vtx‘𝐻) = (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩)
9		ushrisomgr.e	. . . . . . . . . . 11 ⊢ 𝐸 = (Edg‘𝐺)
10	9	fvexi 6451	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
11		id 22	. . . . . . . . . . 11 ⊢ (𝐸 ∈ V → 𝐸 ∈ V)
12	11	resiexd 6741	. . . . . . . . . 10 ⊢ (𝐸 ∈ V → ( I ↾ 𝐸) ∈ V)
13	10, 12	ax-mp 5	. . . . . . . . 9 ⊢ ( I ↾ 𝐸) ∈ V
14	2, 13	pm3.2i 464	. . . . . . . 8 ⊢ (𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V)
15		opvtxfv 26309	. . . . . . . 8 ⊢ ((𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V) → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
16	14, 15	mp1i 13	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → (Vtx‘⟨𝑉, ( I ↾ 𝐸)⟩) = 𝑉)
17	8, 16	syl5eq 2873	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → (Vtx‘𝐻) = 𝑉)
18	17	f1oeq3d 6379	. . . . 5 ⊢ (𝐺 ∈ USHGraph → (( I ↾ 𝑉):𝑉–1-1-onto→(Vtx‘𝐻) ↔ ( I ↾ 𝑉):𝑉–1-1-onto→𝑉))
19	6, 18	mpbird 249	. . . 4 ⊢ (𝐺 ∈ USHGraph → ( I ↾ 𝑉):𝑉–1-1-onto→(Vtx‘𝐻))
20		fvexd 6452	. . . . 5 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺) ∈ V)
21		eqid 2825	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
22	1, 21	ushgrf 26368	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 𝑉 ∖ {∅}))
23		f1f1orn 6393	. . . . . . . 8 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 𝑉 ∖ {∅}) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→ran (iEdg‘𝐺))
24	22, 23	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→ran (iEdg‘𝐺))
25	7	fveq2i 6440	. . . . . . . . . . 11 ⊢ (iEdg‘𝐻) = (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩)
26	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USHGraph → 𝐸 ∈ V)
27	26	resiexd 6741	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → ( I ↾ 𝐸) ∈ V)
28		opiedgfv 26312	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ V ∧ ( I ↾ 𝐸) ∈ V) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
29	2, 27, 28	sylancr 581	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USHGraph → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
30	25, 29	syl5eq 2873	. . . . . . . . . 10 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐻) = ( I ↾ 𝐸))
31	30	dmeqd 5562	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → dom (iEdg‘𝐻) = dom ( I ↾ 𝐸))
32		dmresi 5704	. . . . . . . . . 10 ⊢ dom ( I ↾ 𝐸) = 𝐸
33	9	a1i 11	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USHGraph → 𝐸 = (Edg‘𝐺))
34		edgval 26354	. . . . . . . . . . 11 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
35	33, 34	syl6eq 2877	. . . . . . . . . 10 ⊢ (𝐺 ∈ USHGraph → 𝐸 = ran (iEdg‘𝐺))
36	32, 35	syl5eq 2873	. . . . . . . . 9 ⊢ (𝐺 ∈ USHGraph → dom ( I ↾ 𝐸) = ran (iEdg‘𝐺))
37	31, 36	eqtrd 2861	. . . . . . . 8 ⊢ (𝐺 ∈ USHGraph → dom (iEdg‘𝐻) = ran (iEdg‘𝐺))
38	37	f1oeq3d 6379	. . . . . . 7 ⊢ (𝐺 ∈ USHGraph → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→ran (iEdg‘𝐺)))
39	24, 38	mpbird 249	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻))
40		ushgruhgr 26374	. . . . . . . . . 10 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
41	1, 21	uhgrss 26369	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ 𝑉)
42	40, 41	sylan 575	. . . . . . . . 9 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ 𝑉)
43		resiima 5725	. . . . . . . . 9 ⊢ (((iEdg‘𝐺)‘𝑖) ⊆ 𝑉 → (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
44	42, 43	syl 17	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
45		f1f 6342	. . . . . . . . . . . . 13 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 𝑉 ∖ {∅}) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
46	22, 45	syl 17	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
47	46	ffund 6286	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USHGraph → Fun (iEdg‘𝐺))
48		fvelrn 6606	. . . . . . . . . . 11 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ ran (iEdg‘𝐺))
49	47, 48	sylan 575	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ ran (iEdg‘𝐺))
50	9, 34	eqtri 2849	. . . . . . . . . 10 ⊢ 𝐸 = ran (iEdg‘𝐺)
51	49, 50	syl6eleqr 2917	. . . . . . . . 9 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ∈ 𝐸)
52		fvresi 6696	. . . . . . . . 9 ⊢ (((iEdg‘𝐺)‘𝑖) ∈ 𝐸 → (( I ↾ 𝐸)‘((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
53	51, 52	syl 17	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝐸)‘((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
54	10	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → 𝐸 ∈ V)
55	54	resiexd 6741	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ( I ↾ 𝐸) ∈ V)
56	2, 55, 28	sylancr 581	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (iEdg‘⟨𝑉, ( I ↾ 𝐸)⟩) = ( I ↾ 𝐸))
57	25, 56	syl5req 2874	. . . . . . . . 9 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ( I ↾ 𝐸) = (iEdg‘𝐻))
58	57	fveq1d 6439	. . . . . . . 8 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝐸)‘((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
59	44, 53, 58	3eqtr2d 2867	. . . . . . 7 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
60	59	ralrimiva 3175	. . . . . 6 ⊢ (𝐺 ∈ USHGraph → ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
61	39, 60	jca 507	. . . . 5 ⊢ (𝐺 ∈ USHGraph → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖))))
62		f1oeq1 6371	. . . . . 6 ⊢ (𝑔 = (iEdg‘𝐺) → (𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)))
63		fveq1 6436	. . . . . . . . 9 ⊢ (𝑔 = (iEdg‘𝐺) → (𝑔‘𝑖) = ((iEdg‘𝐺)‘𝑖))
64	63	fveq2d 6441	. . . . . . . 8 ⊢ (𝑔 = (iEdg‘𝐺) → ((iEdg‘𝐻)‘(𝑔‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))
65	64	eqeq2d 2835	. . . . . . 7 ⊢ (𝑔 = (iEdg‘𝐺) → ((( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)) ↔ (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖))))
66	65	ralbidv 3195	. . . . . 6 ⊢ (𝑔 = (iEdg‘𝐺) → (∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)) ↔ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖))))
67	62, 66	anbi12d 624	. . . . 5 ⊢ (𝑔 = (iEdg‘𝐺) → ((𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖))) ↔ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((iEdg‘𝐺)‘𝑖)))))
68	20, 61, 67	elabd 3573	. . . 4 ⊢ (𝐺 ∈ USHGraph → ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖))))
69	19, 68	jca 507	. . 3 ⊢ (𝐺 ∈ USHGraph → (( I ↾ 𝑉):𝑉–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)))))
70		f1oeq1 6371	. . . 4 ⊢ (𝑓 = ( I ↾ 𝑉) → (𝑓:𝑉–1-1-onto→(Vtx‘𝐻) ↔ ( I ↾ 𝑉):𝑉–1-1-onto→(Vtx‘𝐻)))
71		imaeq1 5706	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ 𝑉) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)))
72	71	eqeq1d 2827	. . . . . . 7 ⊢ (𝑓 = ( I ↾ 𝑉) → ((𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)) ↔ (( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖))))
73	72	ralbidv 3195	. . . . . 6 ⊢ (𝑓 = ( I ↾ 𝑉) → (∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)) ↔ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖))))
74	73	anbi2d 622	. . . . 5 ⊢ (𝑓 = ( I ↾ 𝑉) → ((𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖))) ↔ (𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)))))
75	74	exbidv 2020	. . . 4 ⊢ (𝑓 = ( I ↾ 𝑉) → (∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖))) ↔ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)))))
76	70, 75	anbi12d 624	. . 3 ⊢ (𝑓 = ( I ↾ 𝑉) → ((𝑓:𝑉–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)))) ↔ (( I ↾ 𝑉):𝑉–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(( I ↾ 𝑉) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖))))))
77	4, 69, 76	elabd 3573	. 2 ⊢ (𝐺 ∈ USHGraph → ∃𝑓(𝑓:𝑉–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖)))))
78		opex 5155	. . . 4 ⊢ ⟨𝑉, ( I ↾ 𝐸)⟩ ∈ V
79	7, 78	eqeltri 2902	. . 3 ⊢ 𝐻 ∈ V
80		eqid 2825	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
81		eqid 2825	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
82	1, 80, 21, 81	isomgr 42559	. . 3 ⊢ ((𝐺 ∈ USHGraph ∧ 𝐻 ∈ V) → (𝐺 IsomGr 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖))))))
83	79, 82	mpan2 682	. 2 ⊢ (𝐺 ∈ USHGraph → (𝐺 IsomGr 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)(𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑔‘𝑖))))))
84	77, 83	mpbird 249	1 ⊢ (𝐺 ∈ USHGraph → 𝐺 IsomGr 𝐻)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656  ∃wex 1878   ∈ wcel 2164  ∀wral 3117  Vcvv 3414   ∖ cdif 3795   ⊆ wss 3798  ∅c0 4146  𝒫 cpw 4380  {csn 4399  ⟨cop 4405   class class class wbr 4875   I cid 5251  dom cdm 5346  ran crn 5347   ↾ cres 5348   “ cima 5349  Fun wfun 6121  ⟶wf 6123  –1-1→wf1 6124  –1-1-onto→wf1o 6126  ‘cfv 6127  Vtxcvtx 26301  iEdgciedg 26302  Edgcedg 26352  UHGraphcuhgr 26361  USHGraphcushgr 26362   IsomGr cisomgr 42555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-1st 7433  df-2nd 7434  df-vtx 26303  df-iedg 26304  df-edg 26353  df-uhgr 26363  df-ushgr 26364  df-isomgr 42557

This theorem is referenced by: (None)