Theorem grlimedgnedg 48161

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 48160). This theorem proves that the analogon (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ 𝐾 ∈ 𝐼)) → (𝐹 “ 𝐾) ∈ 𝐸) of grimedgi 47966 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 7353	. . . 4 ⊢ (𝑔 = (5 gPetersenGr 1) → (𝑔 GraphLocIso ℎ) = ((5 gPetersenGr 1) GraphLocIso ℎ))
2		fveq2 6822	. . . . 5 ⊢ (𝑔 = (5 gPetersenGr 1) → (Vtx‘𝑔) = (Vtx‘(5 gPetersenGr 1)))
3		fveq2 6822	. . . . . . . 8 ⊢ (𝑔 = (5 gPetersenGr 1) → (Edg‘𝑔) = (Edg‘(5 gPetersenGr 1)))
4	3	eleq2d 2817	. . . . . . 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 3313	. . . . 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 3313	. . . 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 3313	. . 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 7354	. . . 4 ⊢ (ℎ = (5 gPetersenGr 2) → ((5 gPetersenGr 1) GraphLocIso ℎ) = ((5 gPetersenGr 1) GraphLocIso (5 gPetersenGr 2)))
10		eqidd 2732	. . . . . . 7 ⊢ (ℎ = (5 gPetersenGr 2) → {(𝑓‘𝑎), (𝑓‘𝑏)} = {(𝑓‘𝑎), (𝑓‘𝑏)})
11		fveq2 6822	. . . . . . 7 ⊢ (ℎ = (5 gPetersenGr 2) → (Edg‘ℎ) = (Edg‘(5 gPetersenGr 2)))
12	10, 11	neleq12d 3037	. . . . . 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 3197	. . . 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 3313	. . 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 12778	. . . 4 ⊢ 5 ∈ (ℤ≥‘3)
17		gpgprismgrusgra 48088	. . . 4 ⊢ (5 ∈ (ℤ≥‘3) → (5 gPetersenGr 1) ∈ USGraph)
18	16, 17	mp1i 13	. . 3 ⊢ (⊤ → (5 gPetersenGr 1) ∈ USGraph)
19		pgjsgr 48122	. . . 4 ⊢ (5 gPetersenGr 2) ∈ USGraph
20	19	a1i 11	. . 3 ⊢ (⊤ → (5 gPetersenGr 2) ∈ USGraph)
21		fveq1 6821	. . . . . . 7 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → (𝑓‘𝑎) = (( I ↾ ({0, 1} × (0..^5)))‘𝑎))
22		fveq1 6821	. . . . . . 7 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → (𝑓‘𝑏) = (( I ↾ ({0, 1} × (0..^5)))‘𝑏))
23	21, 22	preq12d 4694	. . . . . 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 2732	. . . . . 6 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → (Edg‘(5 gPetersenGr 2)) = (Edg‘(5 gPetersenGr 2)))
25	23, 24	neleq12d 3037	. . . . 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 4686	. . . . . 6 ⊢ (𝑎 = ⟨1, 0⟩ → {𝑎, 𝑏} = {⟨1, 0⟩, 𝑏})
28	27	eleq1d 2816	. . . . 5 ⊢ (𝑎 = ⟨1, 0⟩ → ({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ↔ {⟨1, 0⟩, 𝑏} ∈ (Edg‘(5 gPetersenGr 1))))
29		fveq2 6822	. . . . . . 7 ⊢ (𝑎 = ⟨1, 0⟩ → (( I ↾ ({0, 1} × (0..^5)))‘𝑎) = (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩))
30	29	preq1d 4692	. . . . . 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 2732	. . . . . 6 ⊢ (𝑎 = ⟨1, 0⟩ → (Edg‘(5 gPetersenGr 2)) = (Edg‘(5 gPetersenGr 2)))
32	30, 31	neleq12d 3037	. . . . 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 4687	. . . . . 6 ⊢ (𝑏 = ⟨1, 1⟩ → {⟨1, 0⟩, 𝑏} = {⟨1, 0⟩, ⟨1, 1⟩})
35	34	eleq1d 2816	. . . . 5 ⊢ (𝑏 = ⟨1, 1⟩ → ({⟨1, 0⟩, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ↔ {⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1))))
36		fveq2 6822	. . . . . . 7 ⊢ (𝑏 = ⟨1, 1⟩ → (( I ↾ ({0, 1} × (0..^5)))‘𝑏) = (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩))
37	36	preq2d 4693	. . . . . 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 2732	. . . . . 6 ⊢ (𝑏 = ⟨1, 1⟩ → (Edg‘(5 gPetersenGr 2)) = (Edg‘(5 gPetersenGr 2)))
39	37, 38	neleq12d 3037	. . . . 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 48123	. . . . 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 11105	. . . . . . . 8 ⊢ 1 ∈ V
44	43	prid2 4716	. . . . . . 7 ⊢ 1 ∈ {0, 1}
45		5nn 12208	. . . . . . . 8 ⊢ 5 ∈ ℕ
46		lbfzo0 13596	. . . . . . . 8 ⊢ (0 ∈ (0..^5) ↔ 5 ∈ ℕ)
47	45, 46	mpbir 231	. . . . . . 7 ⊢ 0 ∈ (0..^5)
48	44, 47	opelxpii 5654	. . . . . 6 ⊢ ⟨1, 0⟩ ∈ ({0, 1} × (0..^5))
49		1elfzo1ceilhalf1 47367	. . . . . . . 8 ⊢ (5 ∈ (ℤ≥‘3) → 1 ∈ (1..^(⌈‘(5 / 2))))
50	16, 49	ax-mp 5	. . . . . . 7 ⊢ 1 ∈ (1..^(⌈‘(5 / 2)))
51		eqid 2731	. . . . . . . 8 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
52		eqid 2731	. . . . . . . 8 ⊢ (0..^5) = (0..^5)
53	51, 52	gpgvtx 48073	. . . . . . 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 2830	. . . . 5 ⊢ ⟨1, 0⟩ ∈ (Vtx‘(5 gPetersenGr 1))
56	55	a1i 11	. . . 4 ⊢ (⊤ → ⟨1, 0⟩ ∈ (Vtx‘(5 gPetersenGr 1)))
57		1nn0 12394	. . . . . . . 8 ⊢ 1 ∈ ℕ0
58	45	nnzi 12493	. . . . . . . 8 ⊢ 5 ∈ ℤ
59		1lt5 12297	. . . . . . . 8 ⊢ 1 < 5
60		elfzo0z 13598	. . . . . . . 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 5654	. . . . . 6 ⊢ ⟨1, 1⟩ ∈ ({0, 1} × (0..^5))
63	62, 54	eleqtrri 2830	. . . . 5 ⊢ ⟨1, 1⟩ ∈ (Vtx‘(5 gPetersenGr 1))
64	63	a1i 11	. . . 4 ⊢ (⊤ → ⟨1, 1⟩ ∈ (Vtx‘(5 gPetersenGr 1)))
65		gpg5edgnedg 48160	. . . . . 6 ⊢ ({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {⟨1, 0⟩, ⟨1, 1⟩} ∉ (Edg‘(5 gPetersenGr 2)))
66		fvresi 7107	. . . . . . . . . 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 7107	. . . . . . . . . 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 4691	. . . . . . . 8 ⊢ {(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩)} = {⟨1, 0⟩, ⟨1, 1⟩}
71		neleq1 3038	. . . . . . . 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 3595	. . 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 3591	. 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 2111   ∉ wnel 3032  ∃wrex 3056  {cpr 4578  ⟨cop 4582   class class class wbr 5091   I cid 5510   × cxp 5614   ↾ cres 5618  ‘cfv 6481  (class class class)co 7346  0cc0 11003  1c1 11004   < clt 11143   / cdiv 11771  ℕcn 12122  2c2 12177  3c3 12178  5c5 12180  ℕ₀cn0 12378  ℤcz 12465  ℤ_≥cuz 12729  ..^cfzo 13551  ⌈cceil 13692  Vtxcvtx 28972  Edgcedg 29023  USGraphcusgr 29125   GraphLocIso cgrlim 48006   gPetersenGr cgpg 48070

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080  ax-pre-sup 11081

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-oadd 8389  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-inf 9327  df-dju 9791  df-card 9829  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-div 11772  df-nn 12123  df-2 12185  df-3 12186  df-4 12187  df-5 12188  df-6 12189  df-7 12190  df-8 12191  df-9 12192  df-n0 12379  df-xnn0 12452  df-z 12466  df-dec 12586  df-uz 12730  df-rp 12888  df-ico 13248  df-fz 13405  df-fzo 13552  df-fl 13693  df-ceil 13694  df-mod 13771  df-seq 13906  df-exp 13966  df-hash 14235  df-cj 15003  df-re 15004  df-im 15005  df-sqrt 15139  df-abs 15140  df-dvds 16161  df-struct 17055  df-slot 17090  df-ndx 17102  df-base 17118  df-edgf 28965  df-vtx 28974  df-iedg 28975  df-edg 29024  df-uhgr 29034  df-ushgr 29035  df-upgr 29058  df-umgr 29059  df-uspgr 29126  df-usgr 29127  df-subgr 29244  df-nbgr 29309  df-clnbgr 47849  df-isubgr 47891  df-grim 47908  df-gric 47911  df-stgr 47982  df-grlim 48008  df-gpg 48071

This theorem is referenced by: (None)