Theorem grlimedgnedg 48607

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 48606). This theorem proves that the analogon (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ 𝐾 ∈ 𝐼)) → (𝐹 “ 𝐾) ∈ 𝐸) of grimedgi 48412 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 7374	. . . 4 ⊢ (𝑔 = (5 gPetersenGr 1) → (𝑔 GraphLocIso ℎ) = ((5 gPetersenGr 1) GraphLocIso ℎ))
2		fveq2 6840	. . . . 5 ⊢ (𝑔 = (5 gPetersenGr 1) → (Vtx‘𝑔) = (Vtx‘(5 gPetersenGr 1)))
3		fveq2 6840	. . . . . . . 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 632	. . . . . 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 3312	. . . . 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 3312	. . . 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 3312	. . 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 7375	. . . 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 6840	. . . . . . 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 631	. . . . 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 3202	. . . 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 3312	. . 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 12833	. . . 4 ⊢ 5 ∈ (ℤ≥‘3)
17		gpgprismgrusgra 48534	. . . 4 ⊢ (5 ∈ (ℤ≥‘3) → (5 gPetersenGr 1) ∈ USGraph)
18	16, 17	mp1i 13	. . 3 ⊢ (⊤ → (5 gPetersenGr 1) ∈ USGraph)
19		pgjsgr 48568	. . . 4 ⊢ (5 gPetersenGr 2) ∈ USGraph
20	19	a1i 11	. . 3 ⊢ (⊤ → (5 gPetersenGr 2) ∈ USGraph)
21		fveq1 6839	. . . . . . 7 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → (𝑓‘𝑎) = (( I ↾ ({0, 1} × (0..^5)))‘𝑎))
22		fveq1 6839	. . . . . . 7 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → (𝑓‘𝑏) = (( I ↾ ({0, 1} × (0..^5)))‘𝑏))
23	21, 22	preq12d 4685	. . . . . 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 631	. . . 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 4677	. . . . . 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 6840	. . . . . . 7 ⊢ (𝑎 = ⟨1, 0⟩ → (( I ↾ ({0, 1} × (0..^5)))‘𝑎) = (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩))
30	29	preq1d 4683	. . . . . 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 633	. . . 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 4678	. . . . . 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 6840	. . . . . . 7 ⊢ (𝑏 = ⟨1, 1⟩ → (( I ↾ ({0, 1} × (0..^5)))‘𝑏) = (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩))
37	36	preq2d 4684	. . . . . 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 633	. . . 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 48569	. . . . 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 11140	. . . . . . . 8 ⊢ 1 ∈ V
44	43	prid2 4707	. . . . . . 7 ⊢ 1 ∈ {0, 1}
45		5nn 12267	. . . . . . . 8 ⊢ 5 ∈ ℕ
46		lbfzo0 13654	. . . . . . . 8 ⊢ (0 ∈ (0..^5) ↔ 5 ∈ ℕ)
47	45, 46	mpbir 231	. . . . . . 7 ⊢ 0 ∈ (0..^5)
48	44, 47	opelxpii 5669	. . . . . 6 ⊢ ⟨1, 0⟩ ∈ ({0, 1} × (0..^5))
49		1elfzo1ceilhalf1 47789	. . . . . . . 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 48519	. . . . . . 7 ⊢ ((5 ∈ ℕ ∧ 1 ∈ (1..^(⌈‘(5 / 2)))) → (Vtx‘(5 gPetersenGr 1)) = ({0, 1} × (0..^5)))
54	45, 50, 53	mp2an 693	. . . . . 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 12453	. . . . . . . 8 ⊢ 1 ∈ ℕ0
58	45	nnzi 12551	. . . . . . . 8 ⊢ 5 ∈ ℤ
59		1lt5 12356	. . . . . . . 8 ⊢ 1 < 5
60		elfzo0z 13656	. . . . . . . 8 ⊢ (1 ∈ (0..^5) ↔ (1 ∈ ℕ0 ∧ 5 ∈ ℤ ∧ 1 < 5))
61	57, 58, 59, 60	mpbir3an 1343	. . . . . . 7 ⊢ 1 ∈ (0..^5)
62	44, 61	opelxpii 5669	. . . . . 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 48606	. . . . . 6 ⊢ ({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {⟨1, 0⟩, ⟨1, 1⟩} ∉ (Edg‘(5 gPetersenGr 2)))
66		fvresi 7128	. . . . . . . . . 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 7128	. . . . . . . . . 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 4682	. . . . . . . 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 624	. . . . . 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 3582	. . 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 3578	. 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 1549	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 1542  ⊤wtru 1543   ∈ wcel 2114   ∉ wnel 3036  ∃wrex 3061  {cpr 4569  ⟨cop 4573   class class class wbr 5085   I cid 5525   × cxp 5629   ↾ cres 5633  ‘cfv 6498  (class class class)co 7367  0cc0 11038  1c1 11039   < clt 11179   / cdiv 11807  ℕcn 12174  2c2 12236  3c3 12237  5c5 12239  ℕ₀cn0 12437  ℤcz 12524  ℤ_≥cuz 12788  ..^cfzo 13608  ⌈cceil 13750  Vtxcvtx 29065  Edgcedg 29116  USGraphcusgr 29218   GraphLocIso cgrlim 48452   gPetersenGr cgpg 48516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-inf 9356  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-xnn0 12511  df-z 12525  df-dec 12645  df-uz 12789  df-rp 12943  df-ico 13304  df-fz 13462  df-fzo 13609  df-fl 13751  df-ceil 13752  df-mod 13829  df-seq 13964  df-exp 14024  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-dvds 16222  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-edgf 29058  df-vtx 29067  df-iedg 29068  df-edg 29117  df-uhgr 29127  df-ushgr 29128  df-upgr 29151  df-umgr 29152  df-uspgr 29219  df-usgr 29220  df-subgr 29337  df-nbgr 29402  df-clnbgr 48295  df-isubgr 48337  df-grim 48354  df-gric 48357  df-stgr 48428  df-grlim 48454  df-gpg 48517

This theorem is referenced by: (None)