Theorem isgrim 48124

Description: An isomorphism of graphs is a bijection between their vertices that preserves adjacency. (Contributed by AV, 19-Apr-2025.)

Hypotheses

Ref	Expression
isgrim.v	⊢ 𝑉 = (Vtx‘𝐺)
isgrim.w	⊢ 𝑊 = (Vtx‘𝐻)
isgrim.e	⊢ 𝐸 = (iEdg‘𝐺)
isgrim.d	⊢ 𝐷 = (iEdg‘𝐻)

Assertion

Ref	Expression
isgrim	⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖))))))

Distinct variable groups:   𝑖,𝐹,𝑗   𝑖,𝐺,𝑗   𝑖,𝐻,𝑗

Allowed substitution hints:   𝐷(𝑖,𝑗)   𝐸(𝑖,𝑗)   𝑉(𝑖,𝑗)   𝑊(𝑖,𝑗)   𝑋(𝑖,𝑗)   𝑌(𝑖,𝑗)   𝑍(𝑖,𝑗)

Proof of Theorem isgrim

Dummy variables 𝑑 𝑓 𝑒 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-grim 48120	. . 3 ⊢ GraphIso = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∃𝑗[(iEdg‘𝑔) / 𝑒][(iEdg‘ℎ) / 𝑑](𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖))))})
2		elex 3461	. . . 4 ⊢ (𝐺 ∈ 𝑋 → 𝐺 ∈ V)
3	2	3ad2ant1 1133	. . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → 𝐺 ∈ V)
4		elex 3461	. . . 4 ⊢ (𝐻 ∈ 𝑌 → 𝐻 ∈ V)
5	4	3ad2ant2 1134	. . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → 𝐻 ∈ V)
6		f1of 6774	. . . . . . 7 ⊢ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → 𝑓:(Vtx‘𝐺)⟶(Vtx‘𝐻))
7		fvex 6847	. . . . . . . 8 ⊢ (Vtx‘𝐻) ∈ V
8		fvex 6847	. . . . . . . 8 ⊢ (Vtx‘𝐺) ∈ V
9	7, 8	elmap 8809	. . . . . . 7 ⊢ (𝑓 ∈ ((Vtx‘𝐻) ↑m (Vtx‘𝐺)) ↔ 𝑓:(Vtx‘𝐺)⟶(Vtx‘𝐻))
10	6, 9	sylibr 234	. . . . . 6 ⊢ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → 𝑓 ∈ ((Vtx‘𝐻) ↑m (Vtx‘𝐺)))
11	10	adantr 480	. . . . 5 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)))) → 𝑓 ∈ ((Vtx‘𝐻) ↑m (Vtx‘𝐺)))
12		ovex 7391	. . . . 5 ⊢ ((Vtx‘𝐻) ↑m (Vtx‘𝐺)) ∈ V
13	11, 12	abex 5271	. . . 4 ⊢ {𝑓 ∣ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖))))} ∈ V
14	13	a1i 11	. . 3 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → {𝑓 ∣ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖))))} ∈ V)
15		eqidd 2737	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → 𝑓 = 𝑓)
16		fveq2 6834	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
17	16	adantr 480	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (Vtx‘𝑔) = (Vtx‘𝐺))
18		fveq2 6834	. . . . . . 7 ⊢ (ℎ = 𝐻 → (Vtx‘ℎ) = (Vtx‘𝐻))
19	18	adantl 481	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (Vtx‘ℎ) = (Vtx‘𝐻))
20	15, 17, 19	f1oeq123d 6768	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ↔ 𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)))
21		fvexd 6849	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (iEdg‘𝑔) ∈ V)
22		fveq2 6834	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
23	22	adantr 480	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (iEdg‘𝑔) = (iEdg‘𝐺))
24		fvexd 6849	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ ℎ = 𝐻) ∧ 𝑒 = (iEdg‘𝐺)) → (iEdg‘ℎ) ∈ V)
25		fveq2 6834	. . . . . . . . . . 11 ⊢ (ℎ = 𝐻 → (iEdg‘ℎ) = (iEdg‘𝐻))
26	25	adantl 481	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (iEdg‘ℎ) = (iEdg‘𝐻))
27	26	adantr 480	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ ℎ = 𝐻) ∧ 𝑒 = (iEdg‘𝐺)) → (iEdg‘ℎ) = (iEdg‘𝐻))
28		eqidd 2737	. . . . . . . . . . . 12 ⊢ ((𝑒 = (iEdg‘𝐺) ∧ 𝑑 = (iEdg‘𝐻)) → 𝑗 = 𝑗)
29		dmeq 5852	. . . . . . . . . . . . 13 ⊢ (𝑒 = (iEdg‘𝐺) → dom 𝑒 = dom (iEdg‘𝐺))
30	29	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑒 = (iEdg‘𝐺) ∧ 𝑑 = (iEdg‘𝐻)) → dom 𝑒 = dom (iEdg‘𝐺))
31		dmeq 5852	. . . . . . . . . . . . 13 ⊢ (𝑑 = (iEdg‘𝐻) → dom 𝑑 = dom (iEdg‘𝐻))
32	31	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑒 = (iEdg‘𝐺) ∧ 𝑑 = (iEdg‘𝐻)) → dom 𝑑 = dom (iEdg‘𝐻))
33	28, 30, 32	f1oeq123d 6768	. . . . . . . . . . 11 ⊢ ((𝑒 = (iEdg‘𝐺) ∧ 𝑑 = (iEdg‘𝐻)) → (𝑗:dom 𝑒–1-1-onto→dom 𝑑 ↔ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)))
34		fveq1 6833	. . . . . . . . . . . . 13 ⊢ (𝑑 = (iEdg‘𝐻) → (𝑑‘(𝑗‘𝑖)) = ((iEdg‘𝐻)‘(𝑗‘𝑖)))
35		fveq1 6833	. . . . . . . . . . . . . 14 ⊢ (𝑒 = (iEdg‘𝐺) → (𝑒‘𝑖) = ((iEdg‘𝐺)‘𝑖))
36	35	imaeq2d 6019	. . . . . . . . . . . . 13 ⊢ (𝑒 = (iEdg‘𝐺) → (𝑓 “ (𝑒‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)))
37	34, 36	eqeqan12rd 2751	. . . . . . . . . . . 12 ⊢ ((𝑒 = (iEdg‘𝐺) ∧ 𝑑 = (iEdg‘𝐻)) → ((𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖)) ↔ ((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖))))
38	30, 37	raleqbidv 3316	. . . . . . . . . . 11 ⊢ ((𝑒 = (iEdg‘𝐺) ∧ 𝑑 = (iEdg‘𝐻)) → (∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖)) ↔ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖))))
39	33, 38	anbi12d 632	. . . . . . . . . 10 ⊢ ((𝑒 = (iEdg‘𝐺) ∧ 𝑑 = (iEdg‘𝐻)) → ((𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖))) ↔ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)))))
40	39	adantll 714	. . . . . . . . 9 ⊢ ((((𝑔 = 𝐺 ∧ ℎ = 𝐻) ∧ 𝑒 = (iEdg‘𝐺)) ∧ 𝑑 = (iEdg‘𝐻)) → ((𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖))) ↔ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)))))
41	24, 27, 40	sbcied2 3785	. . . . . . . 8 ⊢ (((𝑔 = 𝐺 ∧ ℎ = 𝐻) ∧ 𝑒 = (iEdg‘𝐺)) → ([(iEdg‘ℎ) / 𝑑](𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖))) ↔ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)))))
42	21, 23, 41	sbcied2 3785	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ([(iEdg‘𝑔) / 𝑒][(iEdg‘ℎ) / 𝑑](𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖))) ↔ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)))))
43		biidd 262	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖))) ↔ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)))))
44	42, 43	bitrd 279	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ([(iEdg‘𝑔) / 𝑒][(iEdg‘ℎ) / 𝑑](𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖))) ↔ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)))))
45	44	exbidv 1922	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → (∃𝑗[(iEdg‘𝑔) / 𝑒][(iEdg‘ℎ) / 𝑑](𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖))) ↔ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)))))
46	20, 45	anbi12d 632	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → ((𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∃𝑗[(iEdg‘𝑔) / 𝑒][(iEdg‘ℎ) / 𝑑](𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖)))) ↔ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖))))))
47	46	abbidv 2802	. . 3 ⊢ ((𝑔 = 𝐺 ∧ ℎ = 𝐻) → {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∃𝑗[(iEdg‘𝑔) / 𝑒][(iEdg‘ℎ) / 𝑑](𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖))))} = {𝑓 ∣ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖))))})
48	1, 3, 5, 14, 47	elovmpod 7602	. 2 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖))))}))
49		id 22	. . . . . 6 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
50		isgrim.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
51	50	eqcomi 2745	. . . . . . 7 ⊢ (Vtx‘𝐺) = 𝑉
52	51	a1i 11	. . . . . 6 ⊢ (𝑓 = 𝐹 → (Vtx‘𝐺) = 𝑉)
53		isgrim.w	. . . . . . . 8 ⊢ 𝑊 = (Vtx‘𝐻)
54	53	eqcomi 2745	. . . . . . 7 ⊢ (Vtx‘𝐻) = 𝑊
55	54	a1i 11	. . . . . 6 ⊢ (𝑓 = 𝐹 → (Vtx‘𝐻) = 𝑊)
56	49, 52, 55	f1oeq123d 6768	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ↔ 𝐹:𝑉–1-1-onto→𝑊))
57		eqidd 2737	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → 𝑗 = 𝑗)
58		isgrim.e	. . . . . . . . . . 11 ⊢ 𝐸 = (iEdg‘𝐺)
59	58	eqcomi 2745	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = 𝐸
60	59	dmeqi 5853	. . . . . . . . 9 ⊢ dom (iEdg‘𝐺) = dom 𝐸
61	60	a1i 11	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → dom (iEdg‘𝐺) = dom 𝐸)
62		isgrim.d	. . . . . . . . . . 11 ⊢ 𝐷 = (iEdg‘𝐻)
63	62	eqcomi 2745	. . . . . . . . . 10 ⊢ (iEdg‘𝐻) = 𝐷
64	63	dmeqi 5853	. . . . . . . . 9 ⊢ dom (iEdg‘𝐻) = dom 𝐷
65	64	a1i 11	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → dom (iEdg‘𝐻) = dom 𝐷)
66	57, 61, 65	f1oeq123d 6768	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ↔ 𝑗:dom 𝐸–1-1-onto→dom 𝐷))
67	63	fveq1i 6835	. . . . . . . . . 10 ⊢ ((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐷‘(𝑗‘𝑖))
68	67	a1i 11	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝐷‘(𝑗‘𝑖)))
69	59	fveq1i 6835	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐺)‘𝑖) = (𝐸‘𝑖)
70	69	a1i 11	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ((iEdg‘𝐺)‘𝑖) = (𝐸‘𝑖))
71	49, 70	imaeq12d 6020	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = (𝐹 “ (𝐸‘𝑖)))
72	68, 71	eqeq12d 2752	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)) ↔ (𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖))))
73	61, 72	raleqbidv 3316	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)) ↔ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖))))
74	66, 73	anbi12d 632	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖))) ↔ (𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖)))))
75	74	exbidv 1922	. . . . 5 ⊢ (𝑓 = 𝐹 → (∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖))) ↔ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖)))))
76	56, 75	anbi12d 632	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖)))) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖))))))
77	76	elabg 3631	. . 3 ⊢ (𝐹 ∈ 𝑍 → (𝐹 ∈ {𝑓 ∣ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖))))} ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖))))))
78	77	3ad2ant3 1135	. 2 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ {𝑓 ∣ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (𝑓 “ ((iEdg‘𝐺)‘𝑖))))} ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖))))))
79	48, 78	bitrd 279	1 ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714  ∀wral 3051  Vcvv 3440  [wsbc 3740  dom cdm 5624   “ cima 5627  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763  Vtxcvtx 29069  iEdgciedg 29070   GraphIso cgrim 48117

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8765  df-grim 48120

This theorem is referenced by:  grimprop  48125  grimidvtxedg  48127  grimcnv  48130  grimco  48131  isuspgrim0  48136  dfgric2  48157