upgrimpths - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > upgrimpths		Structured version Visualization version GIF version

Theorem upgrimpths 49152

Description: Graph isomorphisms between simple pseudographs map paths onto paths. (Contributed by AV, 31-Oct-2025.)

**Hypotheses**
Ref	Expression
upgrimwlk.i	⊢ 𝐼 = (iEdg‘𝐺)
upgrimwlk.j	⊢ 𝐽 = (iEdg‘𝐻)
upgrimwlk.g	⊢ (𝜑 → 𝐺 ∈ USPGraph)
upgrimwlk.h	⊢ (𝜑 → 𝐻 ∈ USPGraph)
upgrimwlk.n	⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))
upgrimwlk.e	⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (^◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))
upgrimpths.p	⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃)

**Assertion**
Ref	Expression
upgrimpths	⊢ (𝜑 → 𝐸(Paths‘𝐻)(𝑁 ∘ 𝑃))

Distinct variable groups: 𝑥,𝐹 𝑥,𝐺 𝑥,𝐼 𝑥,𝐽 𝑥,𝑃 𝜑,𝑥 𝑥,𝐸 𝑥,𝑁 Allowed substitution hint: 𝐻(𝑥)

Proof of Theorem upgrimpths Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgrimwlk.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
2		upgrimwlk.j	. . . 4 ⊢ 𝐽 = (iEdg‘𝐻)
3		upgrimwlk.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ USPGraph)
4		upgrimwlk.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ USPGraph)
5		upgrimwlk.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))
6		upgrimwlk.e	. . . 4 ⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (^◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))
7		upgrimpths.p	. . . . 5 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃)
8		pthistrl 30323	. . . . 5 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)
9	7, 8	syl 18	. . . 4 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)
10	1, 2, 3, 4, 5, 6, 9	upgrimtrls 49149	. . 3 ⊢ (𝜑 → 𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃))
11	1, 2, 3, 4, 5, 6, 7	upgrimpthslem1 49150	. . 3 ⊢ (𝜑 → Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
12		pthiswlk 30325	. . . . . . . . . . . 12 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
13	1	wlkf 30210	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
14	7, 12, 13	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
15		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
16	15	wlkp 30212	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
17	7, 12, 16	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
18	1, 2, 3, 4, 5, 6, 14, 17	upgrimwlklem4 49143	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∘ 𝑃):(0...(♯‘𝐸))⟶(Vtx‘𝐻))
19	18	ffnd 6721	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 ∘ 𝑃) Fn (0...(♯‘𝐸)))
20	1, 2, 3, 4, 5, 6, 14	upgrimwlklem1 49140	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐸) = (♯‘𝐹))
21		wlkcl 30211	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
22	7, 12, 21	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ₀)
23	20, 22	eqeltrd 2864	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐸) ∈ ℕ₀)
24		0elfz 13771	. . . . . . . . . 10 ⊢ ((♯‘𝐸) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐸)))
25	23, 24	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0...(♯‘𝐸)))
26		nn0fz0 13772	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (0...(♯‘𝐹)))
27	22, 26	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
28	20	oveq2d 7446	. . . . . . . . . 10 ⊢ (𝜑 → (0...(♯‘𝐸)) = (0...(♯‘𝐹)))
29	27, 28	eleqtrrd 2867	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐹) ∈ (0...(♯‘𝐸)))
30		fnimapr 6979	. . . . . . . . 9 ⊢ (((𝑁 ∘ 𝑃) Fn (0...(♯‘𝐸)) ∧ 0 ∈ (0...(♯‘𝐸)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐸))) → ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) = {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))})
31	19, 25, 29, 30	syl3anc 1398	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) = {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))})
32	31	eleq2d 2850	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ↔ 𝑥 ∈ {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))}))
33		vex 3458	. . . . . . . 8 ⊢ 𝑥 ∈ V
34	33	elpr 4609	. . . . . . 7 ⊢ (𝑥 ∈ {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))} ↔ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
35	32, 34	bitrdi 290	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ↔ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))))
36	1, 2, 3, 4, 5, 6, 7	upgrimpthslem2 49151	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → (¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0) ∧ ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
37	36	simpld 500	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0))
38		eqeq2 2775	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘0) → (((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0)))
39	38	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘0) → (¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0)))
40	37, 39	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → (𝑥 = ((𝑁 ∘ 𝑃)‘0) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
41	36	simprd 501	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))
42		eqeq2 2775	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) → (((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
43	42	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) → (¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
44	41, 43	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → (𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
45	40, 44	jaod 873	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ((𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
46	45	impancom 457	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → (𝑦 ∈ (1..^(♯‘𝐹)) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
47	46	imp 412	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥)
48	47	nrexdv 3161	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → ¬ ∃𝑦 ∈ (1..^(♯‘𝐹))((𝑁 ∘ 𝑃)‘𝑦) = 𝑥)
49	20	eqcomd 2769	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝐹) = (♯‘𝐸))
50	49	oveq2d 7446	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...(♯‘𝐹)) = (0...(♯‘𝐸)))
51	50	feq2d 6704	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 ∘ 𝑃):(0...(♯‘𝐹))⟶(Vtx‘𝐻) ↔ (𝑁 ∘ 𝑃):(0...(♯‘𝐸))⟶(Vtx‘𝐻)))
52	18, 51	mpbird 260	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 ∘ 𝑃):(0...(♯‘𝐹))⟶(Vtx‘𝐻))
53	52	ffnd 6721	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∘ 𝑃) Fn (0...(♯‘𝐹)))
54	53	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → (𝑁 ∘ 𝑃) Fn (0...(♯‘𝐹)))
55		fzo0ss1 13837	. . . . . . . . . . 11 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
56		fzossfz 13826	. . . . . . . . . . 11 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
57	55, 56	sstri 3922	. . . . . . . . . 10 ⊢ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
58	57	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹)))
59	54, 58	fvelimabd 6969	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → (𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))) ↔ ∃𝑦 ∈ (1..^(♯‘𝐹))((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
60	48, 59	mtbird 328	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
61	60	ex 418	. . . . . 6 ⊢ (𝜑 → ((𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))) → ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))))
62	35, 61	sylbid 243	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) → ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))))
63	62	ralrimiv 3157	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
64		disj 4406	. . . 4 ⊢ ((((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅ ↔ ∀𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
65	63, 64	sylibr 237	. . 3 ⊢ (𝜑 → (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅)
66	20	oveq2d 7446	. . . . . . 7 ⊢ (𝜑 → (1..^(♯‘𝐸)) = (1..^(♯‘𝐹)))
67	66	reseq2d 5975	. . . . . 6 ⊢ (𝜑 → ((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) = ((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
68	67	cnveqd 5858	. . . . 5 ⊢ (𝜑 → ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) = ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
69	68	funeqd 6570	. . . 4 ⊢ (𝜑 → (Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) ↔ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹)))))
70		preq2 4695	. . . . . . . 8 ⊢ ((♯‘𝐸) = (♯‘𝐹) → {0, (♯‘𝐸)} = {0, (♯‘𝐹)})
71	70	imaeq2d 6058	. . . . . . 7 ⊢ ((♯‘𝐸) = (♯‘𝐹) → ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) = ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}))
72		oveq2 7438	. . . . . . . 8 ⊢ ((♯‘𝐸) = (♯‘𝐹) → (1..^(♯‘𝐸)) = (1..^(♯‘𝐹)))
73	72	imaeq2d 6058	. . . . . . 7 ⊢ ((♯‘𝐸) = (♯‘𝐹) → ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸))) = ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
74	71, 73	ineq12d 4121	. . . . . 6 ⊢ ((♯‘𝐸) = (♯‘𝐹) → (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))))
75	74	eqeq1d 2765	. . . . 5 ⊢ ((♯‘𝐸) = (♯‘𝐹) → ((((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = ∅ ↔ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅))
76	20, 75	syl 18	. . . 4 ⊢ (𝜑 → ((((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = ∅ ↔ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅))
77	69, 76	3anbi23d 1467	. . 3 ⊢ (𝜑 → ((𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃) ∧ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) ∧ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = ∅) ↔ (𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃) ∧ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))) ∧ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅)))
78	10, 11, 65, 77	mpbir3and 1361	. 2 ⊢ (𝜑 → (𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃) ∧ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) ∧ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = ∅))
79		ispth 30321	. 2 ⊢ (𝐸(Paths‘𝐻)(𝑁 ∘ 𝑃) ↔ (𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃) ∧ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) ∧ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = ∅))
80	78, 79	sylibr 237	1 ⊢ (𝜑 → 𝐸(Paths‘𝐻)(𝑁 ∘ 𝑃))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1569 ∈ wcel 2146 ∀wral 3080 ∃wrex 3090 ⊆ wss 3900 ∩ cin 4092 ∅c0 4281 {cpr 4590 class class class wbr 5106 ↦ cmpt 5189 ^◡ccnv 5654 dom cdm 5655 ↾ cres 5657 “ cima 5658 ∘ ccom 5659 Fun wfun 6541 Fn wfn 6542 ⟶wf 6543 ‘cfv 6547 (class class class)co 7430 0cc0 11213 1c1 11214 ℕ₀cn0 12619 ...cfz 13652 ..^cfzo 13801 ♯chash 14487 Word cword 14671 Vtxcvtx 29589 iEdgciedg 29590 USPGraphcuspgr 29744 Walkscwlks 30192 Trailsctrls 30288 Pathscpths 30310 GraphIso cgrim 49118

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 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-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-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 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7888 df-1st 8011 df-2nd 8012 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-2o 8491 df-oadd 8494 df-er 8731 df-map 8863 df-pm 8864 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-dju 9995 df-card 10033 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-nn 12349 df-2 12418 df-n0 12620 df-xnn0 12693 df-z 12707 df-uz 12979 df-fz 13653 df-fzo 13802 df-hash 14488 df-word 14672 df-edg 29641 df-uhgr 29651 df-upgr 29675 df-uspgr 29746 df-wlks 30195 df-trls 30290 df-pths 30314 df-grim 49121

This theorem is used by: upgrimcycls 49154

W3C validator