Theorem grlimedgnedg 49434

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 49433). This theorem proves that the analogon (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ 𝐾 ∈ 𝐼)) → (𝐹 “ 𝐾) ∈ 𝐸) of grimedgi 49239 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 7450	. . . 4 ⊢ (𝑔 = (5 gPetersenGr 1) → (𝑔 GraphLocIso ℎ) = ((5 gPetersenGr 1) GraphLocIso ℎ))
2		fveq2 6908	. . . . 5 ⊢ (𝑔 = (5 gPetersenGr 1) → (Vtx‘𝑔) = (Vtx‘(5 gPetersenGr 1)))
3		fveq2 6908	. . . . . . . 8 ⊢ (𝑔 = (5 gPetersenGr 1) → (Edg‘𝑔) = (Edg‘(5 gPetersenGr 1)))
4	3	eleq2d 2850	. . . . . . 7 ⊢ (𝑔 = (5 gPetersenGr 1) → ({𝑎, 𝑏} ∈ (Edg‘𝑔) ↔ {𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1))))
5	4	anbi1d 643	. . . . . 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 3339	. . . . 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 3339	. . . 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 3339	. . 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 7451	. . . 4 ⊢ (ℎ = (5 gPetersenGr 2) → ((5 gPetersenGr 1) GraphLocIso ℎ) = ((5 gPetersenGr 1) GraphLocIso (5 gPetersenGr 2)))
10		eqidd 2764	. . . . . . 7 ⊢ (ℎ = (5 gPetersenGr 2) → {(𝑓‘𝑎), (𝑓‘𝑏)} = {(𝑓‘𝑎), (𝑓‘𝑏)})
11		fveq2 6908	. . . . . . 7 ⊢ (ℎ = (5 gPetersenGr 2) → (Edg‘ℎ) = (Edg‘(5 gPetersenGr 2)))
12	10, 11	neleq12d 3070	. . . . . 6 ⊢ (ℎ = (5 gPetersenGr 2) → ({(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ) ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘(5 gPetersenGr 2))))
13	12	anbi2d 642	. . . . 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 3231	. . . 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 3339	. . 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 13037	. . . 4 ⊢ 5 ∈ (ℤ≥‘3)
17		gpgprismgrusgra 49361	. . . 4 ⊢ (5 ∈ (ℤ≥‘3) → (5 gPetersenGr 1) ∈ USGraph)
18	16, 17	mp1i 14	. . 3 ⊢ (⊤ → (5 gPetersenGr 1) ∈ USGraph)
19		pgjsgr 49395	. . . 4 ⊢ (5 gPetersenGr 2) ∈ USGraph
20	19	a1i 11	. . 3 ⊢ (⊤ → (5 gPetersenGr 2) ∈ USGraph)
21		fveq1 6907	. . . . . . 7 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → (𝑓‘𝑎) = (( I ↾ ({0, 1} × (0..^5)))‘𝑎))
22		fveq1 6907	. . . . . . 7 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → (𝑓‘𝑏) = (( I ↾ ({0, 1} × (0..^5)))‘𝑏))
23	21, 22	preq12d 4702	. . . . . 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 2764	. . . . . 6 ⊢ (𝑓 = ( I ↾ ({0, 1} × (0..^5))) → (Edg‘(5 gPetersenGr 2)) = (Edg‘(5 gPetersenGr 2)))
25	23, 24	neleq12d 3070	. . . . 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 642	. . . 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 4694	. . . . . 6 ⊢ (𝑎 = ⟨1, 0⟩ → {𝑎, 𝑏} = {⟨1, 0⟩, 𝑏})
28	27	eleq1d 2849	. . . . 5 ⊢ (𝑎 = ⟨1, 0⟩ → ({𝑎, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ↔ {⟨1, 0⟩, 𝑏} ∈ (Edg‘(5 gPetersenGr 1))))
29		fveq2 6908	. . . . . . 7 ⊢ (𝑎 = ⟨1, 0⟩ → (( I ↾ ({0, 1} × (0..^5)))‘𝑎) = (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩))
30	29	preq1d 4700	. . . . . 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 2764	. . . . . 6 ⊢ (𝑎 = ⟨1, 0⟩ → (Edg‘(5 gPetersenGr 2)) = (Edg‘(5 gPetersenGr 2)))
32	30, 31	neleq12d 3070	. . . . 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 644	. . . 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 4695	. . . . . 6 ⊢ (𝑏 = ⟨1, 1⟩ → {⟨1, 0⟩, 𝑏} = {⟨1, 0⟩, ⟨1, 1⟩})
35	34	eleq1d 2849	. . . . 5 ⊢ (𝑏 = ⟨1, 1⟩ → ({⟨1, 0⟩, 𝑏} ∈ (Edg‘(5 gPetersenGr 1)) ↔ {⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1))))
36		fveq2 6908	. . . . . . 7 ⊢ (𝑏 = ⟨1, 1⟩ → (( I ↾ ({0, 1} × (0..^5)))‘𝑏) = (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩))
37	36	preq2d 4701	. . . . . 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 2764	. . . . . 6 ⊢ (𝑏 = ⟨1, 1⟩ → (Edg‘(5 gPetersenGr 2)) = (Edg‘(5 gPetersenGr 2)))
39	37, 38	neleq12d 3070	. . . . 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 644	. . . 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 49396	. . . . 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		1elpr01 11331	. . . . . . 7 ⊢ 1 ∈ {0, 1}
44		5nn 12456	. . . . . . . 8 ⊢ 5 ∈ ℕ
45		lbfzo0 13861	. . . . . . . 8 ⊢ (0 ∈ (0..^5) ↔ 5 ∈ ℕ)
46	44, 45	mpbir 234	. . . . . . 7 ⊢ 0 ∈ (0..^5)
47	43, 46	opelxpii 5693	. . . . . 6 ⊢ ⟨1, 0⟩ ∈ ({0, 1} × (0..^5))
48		1elfzo1ceilhalf1 48616	. . . . . . . 8 ⊢ (5 ∈ (ℤ≥‘3) → 1 ∈ (1..^(⌈‘(5 / 2))))
49	16, 48	ax-mp 5	. . . . . . 7 ⊢ 1 ∈ (1..^(⌈‘(5 / 2)))
50		eqid 2763	. . . . . . . 8 ⊢ (1..^(⌈‘(5 / 2))) = (1..^(⌈‘(5 / 2)))
51		eqid 2763	. . . . . . . 8 ⊢ (0..^5) = (0..^5)
52	50, 51	gpgvtx 49346	. . . . . . 7 ⊢ ((5 ∈ ℕ ∧ 1 ∈ (1..^(⌈‘(5 / 2)))) → (Vtx‘(5 gPetersenGr 1)) = ({0, 1} × (0..^5)))
53	44, 49, 52	mp2an 705	. . . . . 6 ⊢ (Vtx‘(5 gPetersenGr 1)) = ({0, 1} × (0..^5))
54	47, 53	eleqtrri 2863	. . . . 5 ⊢ ⟨1, 0⟩ ∈ (Vtx‘(5 gPetersenGr 1))
55	54	a1i 11	. . . 4 ⊢ (⊤ → ⟨1, 0⟩ ∈ (Vtx‘(5 gPetersenGr 1)))
56		1nn0 12649	. . . . . . . 8 ⊢ 1 ∈ ℕ0
57	44	nnzi 12747	. . . . . . . 8 ⊢ 5 ∈ ℤ
58		1lt5 12552	. . . . . . . 8 ⊢ 1 < 5
59		elfzo0z 13863	. . . . . . . 8 ⊢ (1 ∈ (0..^5) ↔ (1 ∈ ℕ0 ∧ 5 ∈ ℤ ∧ 1 < 5))
60	56, 57, 58, 59	mpbir3an 1360	. . . . . . 7 ⊢ 1 ∈ (0..^5)
61	43, 60	opelxpii 5693	. . . . . 6 ⊢ ⟨1, 1⟩ ∈ ({0, 1} × (0..^5))
62	61, 53	eleqtrri 2863	. . . . 5 ⊢ ⟨1, 1⟩ ∈ (Vtx‘(5 gPetersenGr 1))
63	62	a1i 11	. . . 4 ⊢ (⊤ → ⟨1, 1⟩ ∈ (Vtx‘(5 gPetersenGr 1)))
64		gpg5edgnedg 49433	. . . . . 6 ⊢ ({⟨1, 0⟩, ⟨1, 1⟩} ∈ (Edg‘(5 gPetersenGr 1)) ∧ {⟨1, 0⟩, ⟨1, 1⟩} ∉ (Edg‘(5 gPetersenGr 2)))
65		fvresi 7201	. . . . . . . . . 10 ⊢ (⟨1, 0⟩ ∈ ({0, 1} × (0..^5)) → (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩) = ⟨1, 0⟩)
66	47, 65	ax-mp 5	. . . . . . . . 9 ⊢ (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩) = ⟨1, 0⟩
67		fvresi 7201	. . . . . . . . . 10 ⊢ (⟨1, 1⟩ ∈ ({0, 1} × (0..^5)) → (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩) = ⟨1, 1⟩)
68	61, 67	ax-mp 5	. . . . . . . . 9 ⊢ (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩) = ⟨1, 1⟩
69	66, 68	preq12i 4699	. . . . . . . 8 ⊢ {(( I ↾ ({0, 1} × (0..^5)))‘⟨1, 0⟩), (( I ↾ ({0, 1} × (0..^5)))‘⟨1, 1⟩)} = {⟨1, 0⟩, ⟨1, 1⟩}
70		neleq1 3071	. . . . . . . 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))))
71	69, 70	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)))
72	71	anbi2i 635	. . . . . 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))))
73	64, 72	mpbir 234	. . . . 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)))
74	73	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))))
75	26, 33, 40, 42, 55, 63, 74	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))))
76	8, 15, 18, 20, 75	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)))
77	76	mptru 1577	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))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ⊤wtru 1571   ∈ wcel 2147   ∉ wnel 3065  ∃wrex 3090  {cpr 4590  ⟨cop 4838   class class class wbr 5106   I cid 5549   × cxp 5653   ↾ cres 5657  ‘cfv 6558  (class class class)co 7443  0cc0 11227  1c1 11228   < clt 11370   / cdiv 12000  ℕcn 12362  2c2 12424  3c3 12425  5c5 12427  ℕ₀cn0 12633  ℤcz 12720  ℤ_≥cuz 12992  ..^cfzo 13815  ⌈cceil 13958  Vtxcvtx 29604  Edgcedg 29655  USGraphcusgr 29760   GraphLocIso cgrlim 49279   gPetersenGr cgpg 49343

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304  ax-pre-sup 11305

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8745  df-map 8877  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-sup 9461  df-inf 9462  df-dju 10009  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-div 12001  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-xnn0 12707  df-z 12721  df-dec 12842  df-uz 12993  df-rp 13148  df-ico 13509  df-fz 13667  df-fzo 13816  df-fl 13959  df-ceil 13960  df-mod 14037  df-seq 14172  df-exp 14232  df-hash 14502  df-cj 15295  df-re 15296  df-im 15297  df-sqrt 15431  df-abs 15432  df-dvds 16452  df-struct 17354  df-slot 17389  df-ndx 17401  df-base 17417  df-edgf 29597  df-vtx 29606  df-iedg 29607  df-edg 29656  df-uhgr 29666  df-ushgr 29667  df-upgr 29690  df-umgr 29691  df-uspgr 29761  df-usgr 29762  df-subgr 29879  df-nbgr 29944  df-clnbgr 49122  df-isubgr 49164  df-grim 49181  df-gric 49184  df-stgr 49255  df-grlim 49281  df-gpg 49344

This theorem is used by: (None)