uspgrimprop - Mathbox for Alexander van der Vekens

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Theorem uspgrimprop 47811

Description: An isomorphism of simple pseudographs is a bijection between their vertices that preserves adjacency, i.e. there is an edge in one graph connecting one or two vertices iff there is an edge in the other graph connecting the vertices which are the images of the vertices. (Contributed by AV, 27-Apr-2025.)

**Hypotheses**
Ref	Expression
isusgrim.v	⊢ 𝑉 = (Vtx‘𝐺)
isusgrim.w	⊢ 𝑊 = (Vtx‘𝐻)
isusgrim.e	⊢ 𝐸 = (Edg‘𝐺)
isusgrim.d	⊢ 𝐷 = (Edg‘𝐻)

**Assertion**
Ref	Expression
uspgrimprop	⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))))

Distinct variable groups: 𝑥,𝐸 𝑥,𝐹 𝑥,𝐷,𝑦 𝑦,𝐸 𝑦,𝐹 𝑥,𝐺,𝑦 𝑥,𝐻,𝑦 𝑥,𝑉,𝑦 𝑥,𝑊,𝑦

Proof of Theorem uspgrimprop Dummy variables 𝑒 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isusgrim.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
2		isusgrim.w	. . . . . 6 ⊢ 𝑊 = (Vtx‘𝐻)
3		isusgrim.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐺)
4		isusgrim.d	. . . . . 6 ⊢ 𝐷 = (Edg‘𝐻)
5	1, 2, 3, 4	isuspgrim0 47810	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)))
6	5	3expa 1117	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)))
7		simprl 771	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) → 𝐹:𝑉–1-1-onto→𝑊)
8		imaeq2 6076	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = {𝑥, 𝑦} → (𝐹 “ 𝑒) = (𝐹 “ {𝑥, 𝑦}))
9		eqidd 2736	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑉–1-1-onto→𝑊 ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)))
10		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑉–1-1-onto→𝑊 ∧ {𝑥, 𝑦} ∈ 𝐸) → {𝑥, 𝑦} ∈ 𝐸)
11		f1ofun 6851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → Fun 𝐹)
12		zfpair2 5439	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑥, 𝑦} ∈ V
13		funimaexg 6654	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun 𝐹 ∧ {𝑥, 𝑦} ∈ V) → (𝐹 “ {𝑥, 𝑦}) ∈ V)
14	11, 12, 13	sylancl 586	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → (𝐹 “ {𝑥, 𝑦}) ∈ V)
15	14	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑉–1-1-onto→𝑊 ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝐹 “ {𝑥, 𝑦}) ∈ V)
16	8, 9, 10, 15	fvmptd4 7040	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝑉–1-1-onto→𝑊 ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))‘{𝑥, 𝑦}) = (𝐹 “ {𝑥, 𝑦}))
17	16	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → ({𝑥, 𝑦} ∈ 𝐸 → ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))‘{𝑥, 𝑦}) = (𝐹 “ {𝑥, 𝑦})))
18	17	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷) → ({𝑥, 𝑦} ∈ 𝐸 → ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))‘{𝑥, 𝑦}) = (𝐹 “ {𝑥, 𝑦})))
19	18	ad2antlr 727	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ({𝑥, 𝑦} ∈ 𝐸 → ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))‘{𝑥, 𝑦}) = (𝐹 “ {𝑥, 𝑦})))
20	19	imp 406	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))‘{𝑥, 𝑦}) = (𝐹 “ {𝑥, 𝑦}))
21		f1of 6849	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷 → (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸⟶𝐷)
22	21	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) → (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸⟶𝐷)
23		ax-1 6	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∉ 𝐷 → (𝐻 ∈ USPGraph → ∅ ∉ 𝐷))
24		nnel 3054	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ ∅ ∉ 𝐷 ↔ ∅ ∈ 𝐷)
25		uspgruhgr 29216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐻 ∈ USPGraph → 𝐻 ∈ UHGraph)
26		uhgredgn0 29160	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐻 ∈ UHGraph ∧ ∅ ∈ (Edg‘𝐻)) → ∅ ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}))
27	25, 26	sylan 580	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐻 ∈ USPGraph ∧ ∅ ∈ (Edg‘𝐻)) → ∅ ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}))
28		neldifsn 4797	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ¬ ∅ ∈ (𝒫 (Vtx‘𝐻) ∖ {∅})
29	28	pm2.21i 119	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅ ∈ (𝒫 (Vtx‘𝐻) ∖ {∅}) → ∅ ∉ 𝐷)
30	27, 29	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐻 ∈ USPGraph ∧ ∅ ∈ (Edg‘𝐻)) → ∅ ∉ 𝐷)
31	30	expcom 413	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ (Edg‘𝐻) → (𝐻 ∈ USPGraph → ∅ ∉ 𝐷))
32	31, 4	eleq2s 2857	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∈ 𝐷 → (𝐻 ∈ USPGraph → ∅ ∉ 𝐷))
33	24, 32	sylbi 217	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∅ ∉ 𝐷 → (𝐻 ∈ USPGraph → ∅ ∉ 𝐷))
34	23, 33	pm2.61i 182	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 ∈ USPGraph → ∅ ∉ 𝐷)
35	34	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) → ∅ ∉ 𝐷)
36	22, 35	jca 511	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) → ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸⟶𝐷 ∧ ∅ ∉ 𝐷))
37	36	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸⟶𝐷 ∧ ∅ ∉ 𝐷))
38		feldmfvelcdm 7106	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸⟶𝐷 ∧ ∅ ∉ 𝐷) → ({𝑥, 𝑦} ∈ 𝐸 ↔ ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))‘{𝑥, 𝑦}) ∈ 𝐷))
39	37, 38	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ({𝑥, 𝑦} ∈ 𝐸 ↔ ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))‘{𝑥, 𝑦}) ∈ 𝐷))
40	39	biimpa 476	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))‘{𝑥, 𝑦}) ∈ 𝐷)
41	20, 40	eqeltrrd 2840	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ {𝑥, 𝑦} ∈ 𝐸) → (𝐹 “ {𝑥, 𝑦}) ∈ 𝐷)
42		imaeq2 6076	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹 “ {𝑥, 𝑦}) → (^◡𝐹 “ 𝑧) = (^◡𝐹 “ (𝐹 “ {𝑥, 𝑦})))
43		imaeq2 6076	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝑦 → (𝐹 “ 𝑒) = (𝐹 “ 𝑦))
44	43	cbvmptv 5261	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = (𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦))
45		f1oeq1 6837	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = (𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)) → ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷 ↔ (𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)):𝐸–1-1-onto→𝐷))
46	44, 45	mp1i 13	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) → ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷 ↔ (𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)):𝐸–1-1-onto→𝐷))
47		imaeq2 6076	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑥 → (𝐹 “ 𝑒) = (𝐹 “ 𝑥))
48	47	cbvmptv 5261	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))
49		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) → 𝐹:𝑉–1-1-onto→𝑊)
50	49	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)):𝐸–1-1-onto→𝐷) → 𝐹:𝑉–1-1-onto→𝑊)
51		uspgruhgr 29216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
52		uhgredgss 29163	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))
53		difss2 4148	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}) → (Edg‘𝐺) ⊆ 𝒫 (Vtx‘𝐺))
54	51, 52, 53	3syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺 ∈ USPGraph → (Edg‘𝐺) ⊆ 𝒫 (Vtx‘𝐺))
55	1	pweqi 4621	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
56	54, 3, 55	3sstr4g 4041	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺 ∈ USPGraph → 𝐸 ⊆ 𝒫 𝑉)
57	56	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐸 ⊆ 𝒫 𝑉)
58	57	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)):𝐸–1-1-onto→𝐷) → 𝐸 ⊆ 𝒫 𝑉)
59		f1ofo 6856	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)):𝐸–1-1-onto→𝐷 → (𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)):𝐸–onto→𝐷)
60	44	rneqi 5951	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = ran (𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦))
61		forn 6824	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)):𝐸–onto→𝐷 → ran (𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)) = 𝐷)
62	60, 61	eqtrid 2787	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)):𝐸–onto→𝐷 → ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = 𝐷)
63	59, 62	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)):𝐸–1-1-onto→𝐷 → ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = 𝐷)
64	63	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)):𝐸–1-1-onto→𝐷) → ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = 𝐷)
65		uhgredgss 29163	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐻 ∈ UHGraph → (Edg‘𝐻) ⊆ (𝒫 (Vtx‘𝐻) ∖ {∅}))
66		difss2 4148	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Edg‘𝐻) ⊆ (𝒫 (Vtx‘𝐻) ∖ {∅}) → (Edg‘𝐻) ⊆ 𝒫 (Vtx‘𝐻))
67	25, 65, 66	3syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐻 ∈ USPGraph → (Edg‘𝐻) ⊆ 𝒫 (Vtx‘𝐻))
68	2	pweqi 4621	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝒫 𝑊 = 𝒫 (Vtx‘𝐻)
69	67, 4, 68	3sstr4g 4041	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐻 ∈ USPGraph → 𝐷 ⊆ 𝒫 𝑊)
70	69	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐷 ⊆ 𝒫 𝑊)
71	70	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)):𝐸–1-1-onto→𝐷) → 𝐷 ⊆ 𝒫 𝑊)
72	64, 71	eqsstrd 4034	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)):𝐸–1-1-onto→𝐷) → ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) ⊆ 𝒫 𝑊)
73	1	fvexi 6921	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑉 ∈ V
74	73	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)):𝐸–1-1-onto→𝐷) → 𝑉 ∈ V)
75	48, 50, 58, 72, 74	mptcnfimad 8010	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)):𝐸–1-1-onto→𝐷) → ^◡(𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = (𝑧 ∈ ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) ↦ (^◡𝐹 “ 𝑧)))
76	75	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) → ((𝑦 ∈ 𝐸 ↦ (𝐹 “ 𝑦)):𝐸–1-1-onto→𝐷 → ^◡(𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = (𝑧 ∈ ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) ↦ (^◡𝐹 “ 𝑧))))
77	46, 76	sylbid 240	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) → ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷 → ^◡(𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = (𝑧 ∈ ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) ↦ (^◡𝐹 “ 𝑧))))
78	77	impr 454	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) → ^◡(𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = (𝑧 ∈ ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) ↦ (^◡𝐹 “ 𝑧)))
79	78	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ (𝐹 “ {𝑥, 𝑦}) ∈ 𝐷) → ^◡(𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = (𝑧 ∈ ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) ↦ (^◡𝐹 “ 𝑧)))
80		f1ofo 6856	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷 → (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–onto→𝐷)
81		forn 6824	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–onto→𝐷 → ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = 𝐷)
82	81	eqcomd 2741	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–onto→𝐷 → 𝐷 = ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)))
83	80, 82	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷 → 𝐷 = ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)))
84	83	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷) → 𝐷 = ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)))
85	84	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → 𝐷 = ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)))
86	85	eleq2d 2825	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝐹 “ {𝑥, 𝑦}) ∈ 𝐷 ↔ (𝐹 “ {𝑥, 𝑦}) ∈ ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))))
87	86	biimpa 476	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ (𝐹 “ {𝑥, 𝑦}) ∈ 𝐷) → (𝐹 “ {𝑥, 𝑦}) ∈ ran (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)))
88		dff1o2 6854	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝑉–1-1-onto→𝑊 ↔ (𝐹 Fn 𝑉 ∧ Fun ^◡𝐹 ∧ ran 𝐹 = 𝑊))
89	88	simp2bi 1145	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → Fun ^◡𝐹)
90	89	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷) → Fun ^◡𝐹)
91	90	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → Fun ^◡𝐹)
92		funimaexg 6654	. . . . . . . . . . . . . 14 ⊢ ((Fun ^◡𝐹 ∧ (𝐹 “ {𝑥, 𝑦}) ∈ 𝐷) → (^◡𝐹 “ (𝐹 “ {𝑥, 𝑦})) ∈ V)
93	91, 92	sylan 580	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ (𝐹 “ {𝑥, 𝑦}) ∈ 𝐷) → (^◡𝐹 “ (𝐹 “ {𝑥, 𝑦})) ∈ V)
94	42, 79, 87, 93	fvmptd4 7040	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ (𝐹 “ {𝑥, 𝑦}) ∈ 𝐷) → (^◡(𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))‘(𝐹 “ {𝑥, 𝑦})) = (^◡𝐹 “ (𝐹 “ {𝑥, 𝑦})))
95		simplrr 778	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)
96		f1ocnvdm 7305	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷 ∧ (𝐹 “ {𝑥, 𝑦}) ∈ 𝐷) → (^◡(𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))‘(𝐹 “ {𝑥, 𝑦})) ∈ 𝐸)
97	95, 96	sylan 580	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ (𝐹 “ {𝑥, 𝑦}) ∈ 𝐷) → (^◡(𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))‘(𝐹 “ {𝑥, 𝑦})) ∈ 𝐸)
98	94, 97	eqeltrrd 2840	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ (𝐹 “ {𝑥, 𝑦}) ∈ 𝐷) → (^◡𝐹 “ (𝐹 “ {𝑥, 𝑦})) ∈ 𝐸)
99		f1of1 6848	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹:𝑉–1-1→𝑊)
100	99	ad2antrl 728	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) → 𝐹:𝑉–1-1→𝑊)
101		prssi 4826	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → {𝑥, 𝑦} ⊆ 𝑉)
102		f1imacnv 6865	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝑉–1-1→𝑊 ∧ {𝑥, 𝑦} ⊆ 𝑉) → (^◡𝐹 “ (𝐹 “ {𝑥, 𝑦})) = {𝑥, 𝑦})
103	100, 101, 102	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (^◡𝐹 “ (𝐹 “ {𝑥, 𝑦})) = {𝑥, 𝑦})
104	103	eqcomd 2741	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → {𝑥, 𝑦} = (^◡𝐹 “ (𝐹 “ {𝑥, 𝑦})))
105	104	eleq1d 2824	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ({𝑥, 𝑦} ∈ 𝐸 ↔ (^◡𝐹 “ (𝐹 “ {𝑥, 𝑦})) ∈ 𝐸))
106	105	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ (𝐹 “ {𝑥, 𝑦}) ∈ 𝐷) → ({𝑥, 𝑦} ∈ 𝐸 ↔ (^◡𝐹 “ (𝐹 “ {𝑥, 𝑦})) ∈ 𝐸))
107	98, 106	mpbird 257	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ (𝐹 “ {𝑥, 𝑦}) ∈ 𝐷) → {𝑥, 𝑦} ∈ 𝐸)
108	41, 107	impbida 801	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ({𝑥, 𝑦} ∈ 𝐸 ↔ (𝐹 “ {𝑥, 𝑦}) ∈ 𝐷))
109		f1ofn 6850	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹 Fn 𝑉)
110	109	ad2antrl 728	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) → 𝐹 Fn 𝑉)
111	110	anim1i 615	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
112		3anass 1094	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ↔ (𝐹 Fn 𝑉 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
113	111, 112	sylibr 234	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
114		fnimapr 6992	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝐹 “ {𝑥, 𝑦}) = {(𝐹‘𝑥), (𝐹‘𝑦)})
115	113, 114	syl 17	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝐹 “ {𝑥, 𝑦}) = {(𝐹‘𝑥), (𝐹‘𝑦)})
116	115	eleq1d 2824	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝐹 “ {𝑥, 𝑦}) ∈ 𝐷 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))
117	108, 116	bitrd 279	. . . . . . . 8 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))
118	117	ralrimivva 3200	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))
119	7, 118	jca 511	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)))
120	119	ex 412	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → ((𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))))
121	120	adantr 480	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → ((𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))))
122	6, 121	sylbid 240	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))))
123	122	syldbl2 841	. 2 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)))
124	123	ex 412	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∉ wnel 3044 ∀wral 3059 Vcvv 3478 ∖ cdif 3960 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 {csn 4631 {cpr 4633 ↦ cmpt 5231 ^◡ccnv 5688 ran crn 5690 “ cima 5692 Fun wfun 6557 Fn wfn 6558 ⟶wf 6559 –1-1→wf1 6560 –onto→wfo 6561 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 Vtxcvtx 29028 Edgcedg 29079 UHGraphcuhgr 29088 USPGraphcuspgr 29180 GraphIso cgrim 47799

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-edg 29080 df-uhgr 29090 df-upgr 29114 df-uspgr 29182 df-grim 47802

This theorem is referenced by: isuspgrim 47813

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