Theorem isuspgrimlem 48515

Description: Lemma for isuspgrim 48516. (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
isuspgrimlem	⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)

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

Allowed substitution hints:   𝑊(𝑥,𝑦)

Proof of Theorem isuspgrimlem

Dummy variables 𝑑 𝑎 𝑏 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uspgrupgr 29437	. . . . . . . . 9 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
2	1	adantr 485	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐺 ∈ UPGraph)
3	2	adantr 485	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) → 𝐺 ∈ UPGraph)
4	3	adantr 485	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → 𝐺 ∈ UPGraph)
5		isusgrim.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
6		isusgrim.e	. . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺)
7	5, 6	upgredg 29396	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑒 = {𝑎, 𝑏})
8	4, 7	sylan 591	. . . . 5 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ 𝑒 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑒 = {𝑎, 𝑏})
9		preq12 4697	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
10	9	eleq1d 2850	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
11		fveq2 6871	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
12	11	adantr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝐹‘𝑥) = (𝐹‘𝑎))
13		fveq2 6871	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑏 → (𝐹‘𝑦) = (𝐹‘𝑏))
14	13	adantl 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝐹‘𝑦) = (𝐹‘𝑏))
15	12, 14	preq12d 4703	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → {(𝐹‘𝑥), (𝐹‘𝑦)} = {(𝐹‘𝑎), (𝐹‘𝑏)})
16	15	eleq1d 2850	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ({(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐷))
17	10, 16	bibi12d 348	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷) ↔ ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐷)))
18	17	rspc2gv 3594	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷) → ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐷)))
19	18	com12 33	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐷)))
20	19	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐷)))
21	20	imp 411	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐷))
22		f1ofn 6811	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹 Fn 𝑉)
23	22	ad3antlr 743	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐹 Fn 𝑉)
24		simprl 782	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ 𝑉)
25		simpr 489	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
26	25	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉)
27		fnimapr 6954	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐹 “ {𝑎, 𝑏}) = {(𝐹‘𝑎), (𝐹‘𝑏)})
28	23, 24, 26, 27	syl3anc 1394	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝐹 “ {𝑎, 𝑏}) = {(𝐹‘𝑎), (𝐹‘𝑏)})
29	28	eqcomd 2771	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → {(𝐹‘𝑎), (𝐹‘𝑏)} = (𝐹 “ {𝑎, 𝑏}))
30	29	eleq1d 2850	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐷 ↔ (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷))
31	21, 30	bitrd 282	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷))
32	31	adantr 485	. . . . . . . . . . . 12 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑒 = {𝑎, 𝑏}) → ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷))
33	32	biimpd 232	. . . . . . . . . . 11 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑒 = {𝑎, 𝑏}) → ({𝑎, 𝑏} ∈ 𝐸 → (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷))
34		eleq1 2853	. . . . . . . . . . . . 13 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑒 ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
35		imaeq2 6049	. . . . . . . . . . . . . 14 ⊢ (𝑒 = {𝑎, 𝑏} → (𝐹 “ 𝑒) = (𝐹 “ {𝑎, 𝑏}))
36	35	eleq1d 2850	. . . . . . . . . . . . 13 ⊢ (𝑒 = {𝑎, 𝑏} → ((𝐹 “ 𝑒) ∈ 𝐷 ↔ (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷))
37	34, 36	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑒 = {𝑎, 𝑏} → ((𝑒 ∈ 𝐸 → (𝐹 “ 𝑒) ∈ 𝐷) ↔ ({𝑎, 𝑏} ∈ 𝐸 → (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷)))
38	37	adantl 486	. . . . . . . . . . 11 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑒 = {𝑎, 𝑏}) → ((𝑒 ∈ 𝐸 → (𝐹 “ 𝑒) ∈ 𝐷) ↔ ({𝑎, 𝑏} ∈ 𝐸 → (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷)))
39	33, 38	mpbird 260	. . . . . . . . . 10 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑒 = {𝑎, 𝑏}) → (𝑒 ∈ 𝐸 → (𝐹 “ 𝑒) ∈ 𝐷))
40	39	exp31 424	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑒 = {𝑎, 𝑏} → (𝑒 ∈ 𝐸 → (𝐹 “ 𝑒) ∈ 𝐷))))
41	40	com23 87	. . . . . . . 8 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → (𝑒 = {𝑎, 𝑏} → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑒 ∈ 𝐸 → (𝐹 “ 𝑒) ∈ 𝐷))))
42	41	com24 96	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → (𝑒 ∈ 𝐸 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑒 = {𝑎, 𝑏} → (𝐹 “ 𝑒) ∈ 𝐷))))
43	42	imp 411	. . . . . 6 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ 𝑒 ∈ 𝐸) → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑒 = {𝑎, 𝑏} → (𝐹 “ 𝑒) ∈ 𝐷)))
44	43	rexlimdvv 3221	. . . . 5 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ 𝑒 ∈ 𝐸) → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑒 = {𝑎, 𝑏} → (𝐹 “ 𝑒) ∈ 𝐷))
45	8, 44	mpd 16	. . . 4 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ 𝑒 ∈ 𝐸) → (𝐹 “ 𝑒) ∈ 𝐷)
46	45	ex 417	. . 3 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → (𝑒 ∈ 𝐸 → (𝐹 “ 𝑒) ∈ 𝐷))
47	46	ralrimiv 3156	. 2 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) ∈ 𝐷)
48		uspgrupgr 29437	. . . . . 6 ⊢ (𝐻 ∈ USPGraph → 𝐻 ∈ UPGraph)
49	48	ad3antlr 743	. . . . 5 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → 𝐻 ∈ UPGraph)
50		isusgrim.w	. . . . . 6 ⊢ 𝑊 = (Vtx‘𝐻)
51		isusgrim.d	. . . . . 6 ⊢ 𝐷 = (Edg‘𝐻)
52	50, 51	upgredg 29396	. . . . 5 ⊢ ((𝐻 ∈ UPGraph ∧ 𝑑 ∈ 𝐷) → ∃𝑎 ∈ 𝑊 ∃𝑏 ∈ 𝑊 𝑑 = {𝑎, 𝑏})
53	49, 52	sylan 591	. . . 4 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ 𝑑 ∈ 𝐷) → ∃𝑎 ∈ 𝑊 ∃𝑏 ∈ 𝑊 𝑑 = {𝑎, 𝑏})
54		f1ofo 6818	. . . . . . . . . . . 12 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹:𝑉–onto→𝑊)
55		foelrn 7092	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑉–onto→𝑊 ∧ 𝑎 ∈ 𝑊) → ∃𝑚 ∈ 𝑉 𝑎 = (𝐹‘𝑚))
56	55	ex 417	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑉–onto→𝑊 → (𝑎 ∈ 𝑊 → ∃𝑚 ∈ 𝑉 𝑎 = (𝐹‘𝑚)))
57		foelrn 7092	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑉–onto→𝑊 ∧ 𝑏 ∈ 𝑊) → ∃𝑛 ∈ 𝑉 𝑏 = (𝐹‘𝑛))
58	57	ex 417	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑉–onto→𝑊 → (𝑏 ∈ 𝑊 → ∃𝑛 ∈ 𝑉 𝑏 = (𝐹‘𝑛)))
59	56, 58	anim12d 620	. . . . . . . . . . . 12 ⊢ (𝐹:𝑉–onto→𝑊 → ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) → (∃𝑚 ∈ 𝑉 𝑎 = (𝐹‘𝑚) ∧ ∃𝑛 ∈ 𝑉 𝑏 = (𝐹‘𝑛))))
60	54, 59	syl 18	. . . . . . . . . . 11 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) → (∃𝑚 ∈ 𝑉 𝑎 = (𝐹‘𝑚) ∧ ∃𝑛 ∈ 𝑉 𝑏 = (𝐹‘𝑛))))
61	60	adantl 486	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) → ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) → (∃𝑚 ∈ 𝑉 𝑎 = (𝐹‘𝑚) ∧ ∃𝑛 ∈ 𝑉 𝑏 = (𝐹‘𝑛))))
62	61	adantr 485	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) → (∃𝑚 ∈ 𝑉 𝑎 = (𝐹‘𝑚) ∧ ∃𝑛 ∈ 𝑉 𝑏 = (𝐹‘𝑛))))
63	62	imp 411	. . . . . . . 8 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)) → (∃𝑚 ∈ 𝑉 𝑎 = (𝐹‘𝑚) ∧ ∃𝑛 ∈ 𝑉 𝑏 = (𝐹‘𝑛)))
64		preq12 4697	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = (𝐹‘𝑚) ∧ 𝑏 = (𝐹‘𝑛)) → {𝑎, 𝑏} = {(𝐹‘𝑚), (𝐹‘𝑛)})
65	64	eqeq2d 2776	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = (𝐹‘𝑚) ∧ 𝑏 = (𝐹‘𝑛)) → (𝑑 = {𝑎, 𝑏} ↔ 𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)}))
66	65	ancoms 463	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = (𝐹‘𝑛) ∧ 𝑎 = (𝐹‘𝑚)) → (𝑑 = {𝑎, 𝑏} ↔ 𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)}))
67	66	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)) ∧ 𝑚 ∈ 𝑉) ∧ 𝑛 ∈ 𝑉) ∧ (𝑏 = (𝐹‘𝑛) ∧ 𝑎 = (𝐹‘𝑚))) → (𝑑 = {𝑎, 𝑏} ↔ 𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)}))
68		preq12 4697	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → {𝑥, 𝑦} = {𝑚, 𝑛})
69	68	eleq1d 2850	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑚, 𝑛} ∈ 𝐸))
70		fveq2 6871	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑚 → (𝐹‘𝑥) = (𝐹‘𝑚))
71	70	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → (𝐹‘𝑥) = (𝐹‘𝑚))
72		fveq2 6871	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑛 → (𝐹‘𝑦) = (𝐹‘𝑛))
73	72	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → (𝐹‘𝑦) = (𝐹‘𝑛))
74	71, 73	preq12d 4703	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → {(𝐹‘𝑥), (𝐹‘𝑦)} = {(𝐹‘𝑚), (𝐹‘𝑛)})
75	74	eleq1d 2850	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → ({(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷))
76	69, 75	bibi12d 348	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → (({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷) ↔ ({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷)))
77	76	rspc2gv 3594	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷) → ({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷)))
78	77	adantl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷) → ({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷)))
79	22	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) → 𝐹 Fn 𝑉)
80	79	anim1i 626	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
81		3anass 1109	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ↔ (𝐹 Fn 𝑉 ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
82	80, 81	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ 𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉))
83		fnimapr 6954	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → (𝐹 “ {𝑚, 𝑛}) = {(𝐹‘𝑚), (𝐹‘𝑛)})
84	82, 83	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → (𝐹 “ {𝑚, 𝑛}) = {(𝐹‘𝑚), (𝐹‘𝑛)})
85	84	eqcomd 2771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ {𝑚, 𝑛}))
86		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ ({𝑚, 𝑛} ∈ 𝐸 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷)) → ({𝑚, 𝑛} ∈ 𝐸 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷))
87		reueq 3703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ({𝑚, 𝑛} ∈ 𝐸 ↔ ∃!𝑒 ∈ 𝐸 𝑒 = {𝑚, 𝑛})
88	87	bilani 509	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → ∃!𝑒 ∈ 𝐸 𝑒 = {𝑚, 𝑛})
89		eqcom 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ({𝑚, 𝑛} = 𝑒 ↔ 𝑒 = {𝑚, 𝑛})
90	89	reubii 3379	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∃!𝑒 ∈ 𝐸 {𝑚, 𝑛} = 𝑒 ↔ ∃!𝑒 ∈ 𝐸 𝑒 = {𝑚, 𝑛})
91	88, 90	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → ∃!𝑒 ∈ 𝐸 {𝑚, 𝑛} = 𝑒)
92		f1of1 6809	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹:𝑉–1-1→𝑊)
93	92	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) → 𝐹:𝑉–1-1→𝑊)
94	93	ad3antrrr 742	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) ∧ 𝑒 ∈ 𝐸) → 𝐹:𝑉–1-1→𝑊)
95		prssi 4782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → {𝑚, 𝑛} ⊆ 𝑉)
96	95	ad3antlr 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) ∧ 𝑒 ∈ 𝐸) → {𝑚, 𝑛} ⊆ 𝑉)
97		uspgruhgr 29443	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
98	97	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐺 ∈ UHGraph)
99	98	ad3antrrr 742	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → 𝐺 ∈ UHGraph)
100	6	eleq2i 2857	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
101	100	biimpi 219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑒 ∈ 𝐸 → 𝑒 ∈ (Edg‘𝐺))
102		edguhgr 29388	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
103	5	pweqi 4574	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
104	102, 103	eleqtrrdi 2876	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 𝑉)
105	99, 101, 104	syl2an 607	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝒫 𝑉)
106	105	elpwid 4567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) ∧ 𝑒 ∈ 𝐸) → 𝑒 ⊆ 𝑉)
107		f1imaeq 7253	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹:𝑉–1-1→𝑊 ∧ ({𝑚, 𝑛} ⊆ 𝑉 ∧ 𝑒 ⊆ 𝑉)) → ((𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒) ↔ {𝑚, 𝑛} = 𝑒))
108	94, 96, 106, 107	syl12anc 849	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) ∧ 𝑒 ∈ 𝐸) → ((𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒) ↔ {𝑚, 𝑛} = 𝑒))
109	108	reubidva 3384	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → (∃!𝑒 ∈ 𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒) ↔ ∃!𝑒 ∈ 𝐸 {𝑚, 𝑛} = 𝑒))
110	91, 109	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → ∃!𝑒 ∈ 𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒))
111	110	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → ({𝑚, 𝑛} ∈ 𝐸 → ∃!𝑒 ∈ 𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒)))
112	111	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ ({𝑚, 𝑛} ∈ 𝐸 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷)) → ({𝑚, 𝑛} ∈ 𝐸 → ∃!𝑒 ∈ 𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒)))
113	86, 112	sylbird 263	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ ({𝑚, 𝑛} ∈ 𝐸 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷)) → ((𝐹 “ {𝑚, 𝑛}) ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒)))
114	113	ex 417	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → (({𝑚, 𝑛} ∈ 𝐸 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷) → ((𝐹 “ {𝑚, 𝑛}) ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒))))
115		eleq1 2853	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ {𝑚, 𝑛}) → ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷))
116	115	bibi2d 345	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ {𝑚, 𝑛}) → (({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷) ↔ ({𝑚, 𝑛} ∈ 𝐸 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷)))
117		eqeq1 2769	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ({(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ {𝑚, 𝑛}) → ({(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒) ↔ (𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒)))
118	117	reubidv 3386	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ {𝑚, 𝑛}) → (∃!𝑒 ∈ 𝐸 {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒) ↔ ∃!𝑒 ∈ 𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒)))
119	115, 118	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ {𝑚, 𝑛}) → (({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒)) ↔ ((𝐹 “ {𝑚, 𝑛}) ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒))))
120	116, 119	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ {𝑚, 𝑛}) → ((({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷) → ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒))) ↔ (({𝑚, 𝑛} ∈ 𝐸 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷) → ((𝐹 “ {𝑚, 𝑛}) ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒)))))
121	114, 120	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → ({(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ {𝑚, 𝑛}) → (({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷) → ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒)))))
122	85, 121	mpd 16	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → (({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷) → ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒))))
123	78, 122	syld 48	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷) → ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒))))
124	123	impancom 456	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒))))
125	124	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)) → ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒))))
126	125	impl 460	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)) ∧ 𝑚 ∈ 𝑉) ∧ 𝑛 ∈ 𝑉) → ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒)))
127		eleq1 2853	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)} → (𝑑 ∈ 𝐷 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷))
128		eqeq1 2769	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)} → (𝑑 = (𝐹 “ 𝑒) ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒)))
129	128	reubidv 3386	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)} → (∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒) ↔ ∃!𝑒 ∈ 𝐸 {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒)))
130	127, 129	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)} → ((𝑑 ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒)) ↔ ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒))))
131	126, 130	syl5ibrcom 250	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)) ∧ 𝑚 ∈ 𝑉) ∧ 𝑛 ∈ 𝑉) → (𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)} → (𝑑 ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))))
132	131	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)) ∧ 𝑚 ∈ 𝑉) ∧ 𝑛 ∈ 𝑉) ∧ (𝑏 = (𝐹‘𝑛) ∧ 𝑎 = (𝐹‘𝑚))) → (𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)} → (𝑑 ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))))
133	67, 132	sylbid 243	. . . . . . . . . . . . 13 ⊢ ((((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)) ∧ 𝑚 ∈ 𝑉) ∧ 𝑛 ∈ 𝑉) ∧ (𝑏 = (𝐹‘𝑛) ∧ 𝑎 = (𝐹‘𝑚))) → (𝑑 = {𝑎, 𝑏} → (𝑑 ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))))
134	133	exp32 425	. . . . . . . . . . . 12 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)) ∧ 𝑚 ∈ 𝑉) ∧ 𝑛 ∈ 𝑉) → (𝑏 = (𝐹‘𝑛) → (𝑎 = (𝐹‘𝑚) → (𝑑 = {𝑎, 𝑏} → (𝑑 ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))))))
135	134	rexlimdva 3166	. . . . . . . . . . 11 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)) ∧ 𝑚 ∈ 𝑉) → (∃𝑛 ∈ 𝑉 𝑏 = (𝐹‘𝑛) → (𝑎 = (𝐹‘𝑚) → (𝑑 = {𝑎, 𝑏} → (𝑑 ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))))))
136	135	com23 87	. . . . . . . . . 10 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)) ∧ 𝑚 ∈ 𝑉) → (𝑎 = (𝐹‘𝑚) → (∃𝑛 ∈ 𝑉 𝑏 = (𝐹‘𝑛) → (𝑑 = {𝑎, 𝑏} → (𝑑 ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))))))
137	136	rexlimdva 3166	. . . . . . . . 9 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)) → (∃𝑚 ∈ 𝑉 𝑎 = (𝐹‘𝑚) → (∃𝑛 ∈ 𝑉 𝑏 = (𝐹‘𝑛) → (𝑑 = {𝑎, 𝑏} → (𝑑 ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))))))
138	137	impd 415	. . . . . . . 8 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)) → ((∃𝑚 ∈ 𝑉 𝑎 = (𝐹‘𝑚) ∧ ∃𝑛 ∈ 𝑉 𝑏 = (𝐹‘𝑛)) → (𝑑 = {𝑎, 𝑏} → (𝑑 ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒)))))
139	63, 138	mpd 16	. . . . . . 7 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)) → (𝑑 = {𝑎, 𝑏} → (𝑑 ∈ 𝐷 → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))))
140	139	com23 87	. . . . . 6 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ (𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊)) → (𝑑 ∈ 𝐷 → (𝑑 = {𝑎, 𝑏} → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))))
141	140	impancom 456	. . . . 5 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ 𝑑 ∈ 𝐷) → ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) → (𝑑 = {𝑎, 𝑏} → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))))
142	141	rexlimdvv 3221	. . . 4 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ 𝑑 ∈ 𝐷) → (∃𝑎 ∈ 𝑊 ∃𝑏 ∈ 𝑊 𝑑 = {𝑎, 𝑏} → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒)))
143	53, 142	mpd 16	. . 3 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) ∧ 𝑑 ∈ 𝐷) → ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))
144	143	ralrimiva 3157	. 2 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → ∀𝑑 ∈ 𝐷 ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒))
145		eqid 2765	. . 3 ⊢ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))
146	145	f1ompt 7096	. 2 ⊢ ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷 ↔ (∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) ∈ 𝐷 ∧ ∀𝑑 ∈ 𝐷 ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒)))
147	47, 144, 146	sylanbrc 594	1 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  ∃!wreu 3368   ⊆ wss 3907  𝒫 cpw 4558  {cpr 4587   ↦ cmpt 5186   “ cima 5655   Fn wfn 6520  –1-1→wf1 6522  –onto→wfo 6523  –1-1-onto→wf1o 6524  ‘cfv 6525  Vtxcvtx 29255  Edgcedg 29306  UHGraphcuhgr 29315  UPGraphcupgr 29339  USPGraphcuspgr 29407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-oadd 8445  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-dju 9875  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-n0 12496  df-xnn0 12569  df-z 12583  df-uz 12854  df-fz 13527  df-hash 14358  df-edg 29307  df-uhgr 29317  df-upgr 29341  df-uspgr 29409

This theorem is referenced by:  isuspgrim  48516