Theorem grlimedgnedg 48373

Description: In general, the image of an edge of a graph by a local isomprphism is not an edge of the other graph, proven by an example (see gpg5edgnedg 48372). This theorem proves that the analogon (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ 𝐾 ∈ 𝐼)) → (𝐹 “ 𝐾) ∈ 𝐸) of grimedgi 48178 for ordinarily isomorphic graphs does not hold in general. (Contributed by AV, 30-Dec-2025.)

Assertion

Ref	Expression
grlimedgnedg	⊢ ∃𝑔 ∈ USGraph ∃ℎ ∈ USGraph ∃𝑓 ∈ (𝑔 GraphLocIso ℎ)∃𝑎 ∈ (Vtx‘𝑔)∃𝑏 ∈ (Vtx‘𝑔)({𝑎, 𝑏} ∈ (Edg‘𝑔) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ))

Distinct variable group:   𝑎,𝑏,𝑓,𝑔,ℎ

Proof of Theorem grlimedgnedg

Step	Hyp	Ref	Expression
1		oveq1 7365	. . . 4 ⊢ (𝑔 = (5 gPetersenGr 1) → (𝑔 GraphLocIso ℎ) = ((5 gPetersenGr 1) GraphLocIso ℎ))
2		fveq2 6834	. . . . 5 ⊢ (𝑔 = (5 gPetersenGr 1) → (Vtx‘𝑔) = (Vtx‘(5 gPetersenGr 1)))
3		fveq2 6834	. . . . . . . 8 ⊢ (𝑔 = (5 gPetersenGr 1) → (Edg‘𝑔) = (Edg‘(5 gPetersenGr 1)))
4	3	eleq2d 2822	. . . . . . 7 ⊢ (𝑔 = (5 gPetersenGr 1) → ({𝑎, 𝑏} ∈ (Edg‘𝑔) ↔ {𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1))))
5	4	anbi1d 631	. . . . . 6 ⊢ (𝑔 = (5 gPetersenGr 1) → (({𝑎, 𝑏} ∈ (Edg‘𝑔) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ)) ↔ ({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ))))
6	2, 5	rexeqbidv 3317	. . . . 5 ⊢ (𝑔 = (5 gPetersenGr 1) → (∃𝑏 ∈ (Vtx‘𝑔)({𝑎, 𝑏} ∈ (Edg‘𝑔) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ)) ↔ ∃𝑏 ∈ (Vtx‘(5 gPetersenGr 1))({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ))))
7	2, 6	rexeqbidv 3317	. . . 4 ⊢ (𝑔 = (5 gPetersenGr 1) → (∃𝑎 ∈ (Vtx‘𝑔)∃𝑏 ∈ (Vtx‘𝑔)({𝑎, 𝑏} ∈ (Edg‘𝑔) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ)) ↔ ∃𝑎 ∈ (Vtx‘(5 gPetersenGr 1))∃𝑏 ∈ (Vtx‘(5 gPetersenGr 1))({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ))))
8	1, 7	rexeqbidv 3317	. . 3 ⊢ (𝑔 = (5 gPetersenGr 1) → (∃𝑓 ∈ (𝑔 GraphLocIso ℎ)∃𝑎 ∈ (Vtx‘𝑔)∃𝑏 ∈ (Vtx‘𝑔)({𝑎, 𝑏} ∈ (Edg‘𝑔) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ)) ↔ ∃𝑓 ∈ ((5 gPetersenGr 1) GraphLocIso ℎ)∃𝑎 ∈ (Vtx‘(5 gPetersenGr 1))∃𝑏 ∈ (Vtx‘(5 gPetersenGr 1))({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ))))
9		oveq2 7366	. . . 4 ⊢ (ℎ = (5 gPetersenGr 2) → ((5 gPetersenGr 1) GraphLocIso ℎ) = ((5 gPetersenGr 1) GraphLocIso (5 gPetersenGr 2)))
10		eqidd 2737	. . . . . . 7 ⊢ (ℎ = (5 gPetersenGr 2) → {(𝑓‘𝑎), (𝑓‘𝑏)} = {(𝑓‘𝑎), (𝑓‘𝑏)})
11		fveq2 6834	. . . . . . 7 ⊢ (ℎ = (5 gPetersenGr 2) → (Edg‘ℎ) = (Edg‘(5 gPetersenGr 2)))
12	10, 11	neleq12d 3041	. . . . . 6 ⊢ (ℎ = (5 gPetersenGr 2) → ({(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ) ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2))))
13	12	anbi2d 630	. . . . 5 ⊢ (ℎ = (5 gPetersenGr 2) → (({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ)) ↔ ({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2)))))
14	13	2rexbidv 3201	. . . 4 ⊢ (ℎ = (5 gPetersenGr 2) → (∃𝑎 ∈ (Vtx‘(5 gPetersenGr 1))∃𝑏 ∈ (Vtx‘(5 gPetersenGr 1))({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ)) ↔ ∃𝑎 ∈ (Vtx‘(5 gPetersenGr 1))∃𝑏 ∈ (Vtx‘(5 gPetersenGr 1))({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2)))))
15	9, 14	rexeqbidv 3317	. . 3 ⊢ (ℎ = (5 gPetersenGr 2) → (∃𝑓 ∈ ((5 gPetersenGr 1) GraphLocIso ℎ)∃𝑎 ∈ (Vtx‘(5 gPetersenGr 1))∃𝑏 ∈ (Vtx‘(5 gPetersenGr 1))({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ)) ↔ ∃𝑓 ∈ ((5 gPetersenGr 1) GraphLocIso (5 gPetersenGr 2))∃𝑎 ∈ (Vtx‘(5 gPetersenGr 1))∃𝑏 ∈ (Vtx‘(5 gPetersenGr 1))({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2)))))
16		5eluz3 12796	. . . 4 ⊢ 5 ∈ (ℤ≥‘3)
17		gpgprismgrusgra 48300	. . . 4 ⊢ (5 ∈ (ℤ≥‘3) → (5 gPetersenGr 1) ∈ USGraph)
18	16, 17	mp1i 13	. . 3 ⊢ (⊤ → (5 gPetersenGr 1) ∈ USGraph)
19		pgjsgr 48334	. . . 4 ⊢ (5 gPetersenGr 2) ∈ USGraph
20	19	a1i 11	. . 3 ⊢ (⊤ → (5 gPetersenGr 2) ∈ USGraph)
21		fveq1 6833	. . . . . . 7 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → (𝑓‘𝑎) = (( I ↾ ({0, 1} × (0..^5)))‘𝑎))
22		fveq1 6833	. . . . . . 7 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → (𝑓‘𝑏) = (( I ↾ ({0, 1} × (0..^5)))‘𝑏))
23	21, 22	preq12d 4698	. . . . . 6 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → {(𝑓‘𝑎), (𝑓‘𝑏)} = {(( I ↾ ({0, 1} × (0..^5)))‘𝑎), (( I ↾ ({0, 1} × (0..^5)))‘𝑏)})
24		eqidd 2737	. . . . . 6 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → (Edg‘(5 gPetersenGr 2)) = (Edg‘(5 gPetersenGr 2)))
25	23, 24	neleq12d 3041	. . . . 5 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → ({(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2)) ↔ {(( I ↾ ({0, 1} × (0..^5)))‘𝑎), (( I ↾ ({0, 1} × (0..^5)))‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2))))
26	25	anbi2d 630	. . . 4 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → (({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2))) ↔ ({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(( I ↾ ({0, 1} × (0..^5)))‘𝑎), (( I ↾ ({0, 1} × (0..^5)))‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2)))))
27		preq1 4690	. . . . . 6 ⊢ (𝑎 = ⟨1, 0⟩ → {𝑎, 𝑏} = {⟨1, 0⟩, 𝑏})
28	27	eleq1d 2821	. . . . 5 ⊢ (𝑎 = ⟨1, 0⟩ → ({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ↔ {⟨1, 0⟩, 𝑏} ∈ (Edg‘(5 gPetersenGr 1))))
29		fveq2 6834	. . . . . . 7 ⊢ (𝑎 = ⟨1, 0⟩ → (( I ↾ ({0, 1} × (0..^5)))‘𝑎) = (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩))
30	29	preq1d 4696	. . . . . 6 ⊢ (𝑎 = ⟨1, 0⟩ → {(( I ↾ ({0, 1} × (0..^5)))‘𝑎), (( I ↾ ({0, 1} × (0..^5)))‘𝑏)} = {(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘𝑏)})
31		eqidd 2737	. . . . . 6 ⊢ (𝑎 = ⟨1, 0⟩ → (Edg‘(5 gPetersenGr 2)) = (Edg‘(5 gPetersenGr 2)))
32	30, 31	neleq12d 3041	. . . . 5 ⊢ (𝑎 = ⟨1, 0⟩ → ({(( I ↾ ({0, 1} × (0..^5)))‘𝑎), (( I ↾ ({0, 1} × (0..^5)))‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2)) ↔ {(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2))))
33	28, 32	anbi12d 632	. . . 4 ⊢ (𝑎 = ⟨1, 0⟩ → (({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(( I ↾ ({0, 1} × (0..^5)))‘𝑎), (( I ↾ ({0, 1} × (0..^5)))‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2))) ↔ ({⟨1, 0⟩, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2)))))
34		preq2 4691	. . . . . 6 ⊢ (𝑏 = ⟨1, 1⟩ → {⟨1, 0⟩, 𝑏} = {⟨1, 0⟩, ⟨1, 1⟩})
35	34	eleq1d 2821	. . . . 5 ⊢ (𝑏 = ⟨1, 1⟩ → ({⟨1, 0⟩, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ↔ {⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1))))
36		fveq2 6834	. . . . . . 7 ⊢ (𝑏 = ⟨1, 1⟩ → (( I ↾ ({0, 1} × (0..^5)))‘𝑏) = (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩))
37	36	preq2d 4697	. . . . . 6 ⊢ (𝑏 = ⟨1, 1⟩ → {(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘𝑏)} = {(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩)})
38		eqidd 2737	. . . . . 6 ⊢ (𝑏 = ⟨1, 1⟩ → (Edg‘(5 gPetersenGr 2)) = (Edg‘(5 gPetersenGr 2)))
39	37, 38	neleq12d 3041	. . . . 5 ⊢ (𝑏 = ⟨1, 1⟩ → ({(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2)) ↔ {(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩)} ∉ (Edg‘(5 gPetersenGr 2))))
40	35, 39	anbi12d 632	. . . 4 ⊢ (𝑏 = ⟨1, 1⟩ → (({⟨1, 0⟩, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2))) ↔ ({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩)} ∉ (Edg‘(5 gPetersenGr 2)))))
41		gpg5grlim 48335	. . . . 5 ⊢ ( I ↾ ({0, 1} × (0..^5))) ∈ ((5 gPetersenGr 1) GraphLocIso (5 gPetersenGr 2))
42	41	a1i 11	. . . 4 ⊢ (⊤ → ( I ↾ ({0, 1} × (0..^5))) ∈ ((5 gPetersenGr 1) GraphLocIso (5 gPetersenGr 2)))
43		1ex 11128	. . . . . . . 8 ⊢ 1 ∈ V
44	43	prid2 4720	. . . . . . 7 ⊢ 1 ∈ {0, 1}
45		5nn 12231	. . . . . . . 8 ⊢ 5 ∈ ℕ
46		lbfzo0 13615	. . . . . . . 8 ⊢ (0 ∈ (0..^5) ↔ 5 ∈ ℕ)
47	45, 46	mpbir 231	. . . . . . 7 ⊢ 0 ∈ (0..^5)
48	44, 47	opelxpii 5662	. . . . . 6 ⊢ ⟨1, 0⟩ ∈ ({0, 1} × (0..^5))
49		1elfzo1ceilhalf1 47579	. . . . . . . 8 ⊢ (5 ∈ (ℤ≥‘3) → 1 ∈ (1..^(⌈‘(5 / 2))))
50	16, 49	ax-mp 5	. . . . . . 7 ⊢ 1 ∈ (1..^(⌈‘(5 / 2)))
51		eqid 2736	. . . . . . . 8 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
52		eqid 2736	. . . . . . . 8 ⊢ (0..^5) = (0..^5)
53	51, 52	gpgvtx 48285	. . . . . . 7 ⊢ ((5 ∈ ℕ ∧ 1 ∈ (1..^(⌈‘(5 / 2)))) → (Vtx‘(5 gPetersenGr 1)) = ({0, 1} × (0..^5)))
54	45, 50, 53	mp2an 692	. . . . . 6 ⊢ (Vtx‘(5 gPetersenGr 1)) = ({0, 1} × (0..^5))
55	48, 54	eleqtrri 2835	. . . . 5 ⊢ ⟨1, 0⟩ ∈ (Vtx‘(5 gPetersenGr 1))
56	55	a1i 11	. . . 4 ⊢ (⊤ → ⟨1, 0⟩ ∈ (Vtx‘(5 gPetersenGr 1)))
57		1nn0 12417	. . . . . . . 8 ⊢ 1 ∈ ℕ0
58	45	nnzi 12515	. . . . . . . 8 ⊢ 5 ∈ ℤ
59		1lt5 12320	. . . . . . . 8 ⊢ 1 < 5
60		elfzo0z 13617	. . . . . . . 8 ⊢ (1 ∈ (0..^5) ↔ (1 ∈ ℕ0 ∧ 5 ∈ ℤ ∧ 1 < 5))
61	57, 58, 59, 60	mpbir3an 1342	. . . . . . 7 ⊢ 1 ∈ (0..^5)
62	44, 61	opelxpii 5662	. . . . . 6 ⊢ ⟨1, 1⟩ ∈ ({0, 1} × (0..^5))
63	62, 54	eleqtrri 2835	. . . . 5 ⊢ ⟨1, 1⟩ ∈ (Vtx‘(5 gPetersenGr 1))
64	63	a1i 11	. . . 4 ⊢ (⊤ → ⟨1, 1⟩ ∈ (Vtx‘(5 gPetersenGr 1)))
65		gpg5edgnedg 48372	. . . . . 6 ⊢ ({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {⟨1, 0⟩, ⟨1, 1⟩} ∉ (Edg‘(5 gPetersenGr 2)))
66		fvresi 7119	. . . . . . . . . 10 ⊢ (⟨1, 0⟩ ∈ ({0, 1} × (0..^5)) → (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩) = ⟨1, 0⟩)
67	48, 66	ax-mp 5	. . . . . . . . 9 ⊢ (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩) = ⟨1, 0⟩
68		fvresi 7119	. . . . . . . . . 10 ⊢ (⟨1, 1⟩ ∈ ({0, 1} × (0..^5)) → (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩) = ⟨1, 1⟩)
69	62, 68	ax-mp 5	. . . . . . . . 9 ⊢ (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩) = ⟨1, 1⟩
70	67, 69	preq12i 4695	. . . . . . . 8 ⊢ {(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩)} = {⟨1, 0⟩, ⟨1, 1⟩}
71		neleq1 3042	. . . . . . . 8 ⊢ ({(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩)} = {⟨1, 0⟩, ⟨1, 1⟩} → ({(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩)} ∉ (Edg‘(5 gPetersenGr 2)) ↔ {⟨1, 0⟩, ⟨1, 1⟩} ∉ (Edg‘(5 gPetersenGr 2))))
72	70, 71	ax-mp 5	. . . . . . 7 ⊢ ({(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩)} ∉ (Edg‘(5 gPetersenGr 2)) ↔ {⟨1, 0⟩, ⟨1, 1⟩} ∉ (Edg‘(5 gPetersenGr 2)))
73	72	anbi2i 623	. . . . . 6 ⊢ (({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩)} ∉ (Edg‘(5 gPetersenGr 2))) ↔ ({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {⟨1, 0⟩, ⟨1, 1⟩} ∉ (Edg‘(5 gPetersenGr 2))))
74	65, 73	mpbir 231	. . . . 5 ⊢ ({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩)} ∉ (Edg‘(5 gPetersenGr 2)))
75	74	a1i 11	. . . 4 ⊢ (⊤ → ({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩)} ∉ (Edg‘(5 gPetersenGr 2))))
76	26, 33, 40, 42, 56, 64, 75	3rspcedvdw 3594	. . 3 ⊢ (⊤ → ∃𝑓 ∈ ((5 gPetersenGr 1) GraphLocIso (5 gPetersenGr 2))∃𝑎 ∈ (Vtx‘(5 gPetersenGr 1))∃𝑏 ∈ (Vtx‘(5 gPetersenGr 1))({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2))))
77	8, 15, 18, 20, 76	2rspcedvdw 3590	. 2 ⊢ (⊤ → ∃𝑔 ∈ USGraph ∃ℎ ∈ USGraph ∃𝑓 ∈ (𝑔 GraphLocIso ℎ)∃𝑎 ∈ (Vtx‘𝑔)∃𝑏 ∈ (Vtx‘𝑔)({𝑎, 𝑏} ∈ (Edg‘𝑔) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ)))
78	77	mptru 1548	1 ⊢ ∃𝑔 ∈ USGraph ∃ℎ ∈ USGraph ∃𝑓 ∈ (𝑔 GraphLocIso ℎ)∃𝑎 ∈ (Vtx‘𝑔)∃𝑏 ∈ (Vtx‘𝑔)({𝑎, 𝑏} ∈ (Edg‘𝑔) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ⊤wtru 1542   ∈ wcel 2113   ∉ wnel 3036  ∃wrex 3060  {cpr 4582  ⟨cop 4586   class class class wbr 5098   I cid 5518   × cxp 5622   ↾ cres 5626  ‘cfv 6492  (class class class)co 7358  0cc0 11026  1c1 11027   < clt 11166   / cdiv 11794  ℕcn 12145  2c2 12200  3c3 12201  5c5 12203  ℕ₀cn0 12401  ℤcz 12488  ℤ_≥cuz 12751  ..^cfzo 13570  ⌈cceil 13711  Vtxcvtx 29069  Edgcedg 29120  USGraphcusgr 29222   GraphLocIso cgrlim 48218   gPetersenGr cgpg 48282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-dju 9813  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-xnn0 12475  df-z 12489  df-dec 12608  df-uz 12752  df-rp 12906  df-ico 13267  df-fz 13424  df-fzo 13571  df-fl 13712  df-ceil 13713  df-mod 13790  df-seq 13925  df-exp 13985  df-hash 14254  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-dvds 16180  df-struct 17074  df-slot 17109  df-ndx 17121  df-base 17137  df-edgf 29062  df-vtx 29071  df-iedg 29072  df-edg 29121  df-uhgr 29131  df-ushgr 29132  df-upgr 29155  df-umgr 29156  df-uspgr 29223  df-usgr 29224  df-subgr 29341  df-nbgr 29406  df-clnbgr 48061  df-isubgr 48103  df-grim 48120  df-gric 48123  df-stgr 48194  df-grlim 48220  df-gpg 48283

This theorem is referenced by: (None)