Theorem grlimedgnedg 48116

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 48115). This theorem proves that the analogon (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ 𝐾 ∈ 𝐼)) → (𝐹 “ 𝐾) ∈ 𝐸) of grimedgi 47921 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 7360	. . . 4 ⊢ (𝑔 = (5 gPetersenGr 1) → (𝑔 GraphLocIso ℎ) = ((5 gPetersenGr 1) GraphLocIso ℎ))
2		fveq2 6826	. . . . 5 ⊢ (𝑔 = (5 gPetersenGr 1) → (Vtx‘𝑔) = (Vtx‘(5 gPetersenGr 1)))
3		fveq2 6826	. . . . . . . 8 ⊢ (𝑔 = (5 gPetersenGr 1) → (Edg‘𝑔) = (Edg‘(5 gPetersenGr 1)))
4	3	eleq2d 2814	. . . . . . 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 3311	. . . . 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 3311	. . . 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 3311	. . 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 7361	. . . 4 ⊢ (ℎ = (5 gPetersenGr 2) → ((5 gPetersenGr 1) GraphLocIso ℎ) = ((5 gPetersenGr 1) GraphLocIso (5 gPetersenGr 2)))
10		eqidd 2730	. . . . . . 7 ⊢ (ℎ = (5 gPetersenGr 2) → {(𝑓‘𝑎), (𝑓‘𝑏)} = {(𝑓‘𝑎), (𝑓‘𝑏)})
11		fveq2 6826	. . . . . . 7 ⊢ (ℎ = (5 gPetersenGr 2) → (Edg‘ℎ) = (Edg‘(5 gPetersenGr 2)))
12	10, 11	neleq12d 3034	. . . . . 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 3194	. . . 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 3311	. . 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 12802	. . . 4 ⊢ 5 ∈ (ℤ≥‘3)
17		gpgprismgrusgra 48043	. . . 4 ⊢ (5 ∈ (ℤ≥‘3) → (5 gPetersenGr 1) ∈ USGraph)
18	16, 17	mp1i 13	. . 3 ⊢ (⊤ → (5 gPetersenGr 1) ∈ USGraph)
19		pgjsgr 48077	. . . 4 ⊢ (5 gPetersenGr 2) ∈ USGraph
20	19	a1i 11	. . 3 ⊢ (⊤ → (5 gPetersenGr 2) ∈ USGraph)
21		fveq1 6825	. . . . . . 7 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → (𝑓‘𝑎) = (( I ↾ ({0, 1} × (0..^5)))‘𝑎))
22		fveq1 6825	. . . . . . 7 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → (𝑓‘𝑏) = (( I ↾ ({0, 1} × (0..^5)))‘𝑏))
23	21, 22	preq12d 4695	. . . . . 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 2730	. . . . . 6 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → (Edg‘(5 gPetersenGr 2)) = (Edg‘(5 gPetersenGr 2)))
25	23, 24	neleq12d 3034	. . . . 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 4687	. . . . . 6 ⊢ (𝑎 = ⟨1, 0⟩ → {𝑎, 𝑏} = {⟨1, 0⟩, 𝑏})
28	27	eleq1d 2813	. . . . 5 ⊢ (𝑎 = ⟨1, 0⟩ → ({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ↔ {⟨1, 0⟩, 𝑏} ∈ (Edg‘(5 gPetersenGr 1))))
29		fveq2 6826	. . . . . . 7 ⊢ (𝑎 = ⟨1, 0⟩ → (( I ↾ ({0, 1} × (0..^5)))‘𝑎) = (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩))
30	29	preq1d 4693	. . . . . 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 2730	. . . . . 6 ⊢ (𝑎 = ⟨1, 0⟩ → (Edg‘(5 gPetersenGr 2)) = (Edg‘(5 gPetersenGr 2)))
32	30, 31	neleq12d 3034	. . . . 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 4688	. . . . . 6 ⊢ (𝑏 = ⟨1, 1⟩ → {⟨1, 0⟩, 𝑏} = {⟨1, 0⟩, ⟨1, 1⟩})
35	34	eleq1d 2813	. . . . 5 ⊢ (𝑏 = ⟨1, 1⟩ → ({⟨1, 0⟩, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ↔ {⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1))))
36		fveq2 6826	. . . . . . 7 ⊢ (𝑏 = ⟨1, 1⟩ → (( I ↾ ({0, 1} × (0..^5)))‘𝑏) = (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩))
37	36	preq2d 4694	. . . . . 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 2730	. . . . . 6 ⊢ (𝑏 = ⟨1, 1⟩ → (Edg‘(5 gPetersenGr 2)) = (Edg‘(5 gPetersenGr 2)))
39	37, 38	neleq12d 3034	. . . . 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 48078	. . . . 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 11130	. . . . . . . 8 ⊢ 1 ∈ V
44	43	prid2 4717	. . . . . . 7 ⊢ 1 ∈ {0, 1}
45		5nn 12232	. . . . . . . 8 ⊢ 5 ∈ ℕ
46		lbfzo0 13620	. . . . . . . 8 ⊢ (0 ∈ (0..^5) ↔ 5 ∈ ℕ)
47	45, 46	mpbir 231	. . . . . . 7 ⊢ 0 ∈ (0..^5)
48	44, 47	opelxpii 5661	. . . . . 6 ⊢ ⟨1, 0⟩ ∈ ({0, 1} × (0..^5))
49		1elfzo1ceilhalf1 47322	. . . . . . . 8 ⊢ (5 ∈ (ℤ≥‘3) → 1 ∈ (1..^(⌈‘(5 / 2))))
50	16, 49	ax-mp 5	. . . . . . 7 ⊢ 1 ∈ (1..^(⌈‘(5 / 2)))
51		eqid 2729	. . . . . . . 8 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
52		eqid 2729	. . . . . . . 8 ⊢ (0..^5) = (0..^5)
53	51, 52	gpgvtx 48028	. . . . . . 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 2827	. . . . 5 ⊢ ⟨1, 0⟩ ∈ (Vtx‘(5 gPetersenGr 1))
56	55	a1i 11	. . . 4 ⊢ (⊤ → ⟨1, 0⟩ ∈ (Vtx‘(5 gPetersenGr 1)))
57		1nn0 12418	. . . . . . . 8 ⊢ 1 ∈ ℕ0
58	45	nnzi 12517	. . . . . . . 8 ⊢ 5 ∈ ℤ
59		1lt5 12321	. . . . . . . 8 ⊢ 1 < 5
60		elfzo0z 13622	. . . . . . . 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 5661	. . . . . 6 ⊢ ⟨1, 1⟩ ∈ ({0, 1} × (0..^5))
63	62, 54	eleqtrri 2827	. . . . 5 ⊢ ⟨1, 1⟩ ∈ (Vtx‘(5 gPetersenGr 1))
64	63	a1i 11	. . . 4 ⊢ (⊤ → ⟨1, 1⟩ ∈ (Vtx‘(5 gPetersenGr 1)))
65		gpg5edgnedg 48115	. . . . . 6 ⊢ ({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {⟨1, 0⟩, ⟨1, 1⟩} ∉ (Edg‘(5 gPetersenGr 2)))
66		fvresi 7113	. . . . . . . . . 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 7113	. . . . . . . . . 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 4692	. . . . . . . 8 ⊢ {(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩)} = {⟨1, 0⟩, ⟨1, 1⟩}
71		neleq1 3035	. . . . . . . 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 3597	. . 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 3593	. 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 1547	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 1540  ⊤wtru 1541   ∈ wcel 2109   ∉ wnel 3029  ∃wrex 3053  {cpr 4581  ⟨cop 4585   class class class wbr 5095   I cid 5517   × cxp 5621   ↾ cres 5625  ‘cfv 6486  (class class class)co 7353  0cc0 11028  1c1 11029   < clt 11168   / cdiv 11795  ℕcn 12146  2c2 12201  3c3 12202  5c5 12204  ℕ₀cn0 12402  ℤcz 12489  ℤ_≥cuz 12753  ..^cfzo 13575  ⌈cceil 13713  Vtxcvtx 28959  Edgcedg 29010  USGraphcusgr 29112   GraphLocIso cgrlim 47961   gPetersenGr cgpg 48025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-inf 9352  df-dju 9816  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-div 11796  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-7 12214  df-8 12215  df-9 12216  df-n0 12403  df-xnn0 12476  df-z 12490  df-dec 12610  df-uz 12754  df-rp 12912  df-ico 13272  df-fz 13429  df-fzo 13576  df-fl 13714  df-ceil 13715  df-mod 13792  df-seq 13927  df-exp 13987  df-hash 14256  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-dvds 16182  df-struct 17076  df-slot 17111  df-ndx 17123  df-base 17139  df-edgf 28952  df-vtx 28961  df-iedg 28962  df-edg 29011  df-uhgr 29021  df-ushgr 29022  df-upgr 29045  df-umgr 29046  df-uspgr 29113  df-usgr 29114  df-subgr 29231  df-nbgr 29296  df-clnbgr 47804  df-isubgr 47846  df-grim 47863  df-gric 47866  df-stgr 47937  df-grlim 47963  df-gpg 48026

This theorem is referenced by: (None)