Theorem isuspgrimlem 48137

Description: Lemma for isuspgrim 48138. (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 29251	. . . . . . . . 9 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
2	1	adantr 480	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐺 ∈ UPGraph)
3	2	adantr 480	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) → 𝐺 ∈ UPGraph)
4	3	adantr 480	. . . . . 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 29210	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑒 = {𝑎, 𝑏})
8	4, 7	sylan 580	. . . . 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 4692	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
10	9	eleq1d 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
11		fveq2 6834	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
12	11	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝐹‘𝑥) = (𝐹‘𝑎))
13		fveq2 6834	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑏 → (𝐹‘𝑦) = (𝐹‘𝑏))
14	13	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝐹‘𝑦) = (𝐹‘𝑏))
15	12, 14	preq12d 4698	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → {(𝐹‘𝑥), (𝐹‘𝑦)} = {(𝐹‘𝑎), (𝐹‘𝑏)})
16	15	eleq1d 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ({(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐷))
17	10, 16	bibi12d 345	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷) ↔ ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐷)))
18	17	rspc2gv 3586	. . . . . . . . . . . . . . . . 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 32	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . 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 6775	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹 Fn 𝑉)
23	22	ad3antlr 731	. . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
26	25	adantl 481	. . . . . . . . . . . . . . . . 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 6917	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐹 “ {𝑎, 𝑏}) = {(𝐹‘𝑎), (𝐹‘𝑏)})
28	23, 24, 26, 27	syl3anc 1373	. . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . 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 279	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 229	. . . . . . . . . . 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 2824	. . . . . . . . . . . . 13 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑒 ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
35		imaeq2 6015	. . . . . . . . . . . . . 14 ⊢ (𝑒 = {𝑎, 𝑏} → (𝐹 “ 𝑒) = (𝐹 “ {𝑎, 𝑏}))
36	35	eleq1d 2821	. . . . . . . . . . . . 13 ⊢ (𝑒 = {𝑎, 𝑏} → ((𝐹 “ 𝑒) ∈ 𝐷 ↔ (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷))
37	34, 36	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑒 = {𝑎, 𝑏} → ((𝑒 ∈ 𝐸 → (𝐹 “ 𝑒) ∈ 𝐷) ↔ ({𝑎, 𝑏} ∈ 𝐸 → (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷)))
38	37	adantl 481	. . . . . . . . . . 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 257	. . . . . . . . . 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 419	. . . . . . . . 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 86	. . . . . . . 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 95	. . . . . . 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 406	. . . . . 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 3192	. . . . 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 15	. . . 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 412	. . 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 3127	. 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 29251	. . . . . 6 ⊢ (𝐻 ∈ USPGraph → 𝐻 ∈ UPGraph)
49	48	ad3antlr 731	. . . . 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 29210	. . . . 5 ⊢ ((𝐻 ∈ UPGraph ∧ 𝑑 ∈ 𝐷) → ∃𝑎 ∈ 𝑊 ∃𝑏 ∈ 𝑊 𝑑 = {𝑎, 𝑏})
53	49, 52	sylan 580	. . . 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 6781	. . . . . . . . . . . 12 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹:𝑉–onto→𝑊)
55		foelrn 7052	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑉–onto→𝑊 ∧ 𝑎 ∈ 𝑊) → ∃𝑚 ∈ 𝑉 𝑎 = (𝐹‘𝑚))
56	55	ex 412	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑉–onto→𝑊 → (𝑎 ∈ 𝑊 → ∃𝑚 ∈ 𝑉 𝑎 = (𝐹‘𝑚)))
57		foelrn 7052	. . . . . . . . . . . . . 14 ⊢ ((𝐹:𝑉–onto→𝑊 ∧ 𝑏 ∈ 𝑊) → ∃𝑛 ∈ 𝑉 𝑏 = (𝐹‘𝑛))
58	57	ex 412	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑉–onto→𝑊 → (𝑏 ∈ 𝑊 → ∃𝑛 ∈ 𝑉 𝑏 = (𝐹‘𝑛)))
59	56, 58	anim12d 609	. . . . . . . . . . . 12 ⊢ (𝐹:𝑉–onto→𝑊 → ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) → (∃𝑚 ∈ 𝑉 𝑎 = (𝐹‘𝑚) ∧ ∃𝑛 ∈ 𝑉 𝑏 = (𝐹‘𝑛))))
60	54, 59	syl 17	. . . . . . . . . . 11 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) → (∃𝑚 ∈ 𝑉 𝑎 = (𝐹‘𝑚) ∧ ∃𝑛 ∈ 𝑉 𝑏 = (𝐹‘𝑛))))
61	60	adantl 481	. . . . . . . . . 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 480	. . . . . . . . 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 406	. . . . . . . 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 4692	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = (𝐹‘𝑚) ∧ 𝑏 = (𝐹‘𝑛)) → {𝑎, 𝑏} = {(𝐹‘𝑚), (𝐹‘𝑛)})
65	64	eqeq2d 2747	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = (𝐹‘𝑚) ∧ 𝑏 = (𝐹‘𝑛)) → (𝑑 = {𝑎, 𝑏} ↔ 𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)}))
66	65	ancoms 458	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = (𝐹‘𝑛) ∧ 𝑎 = (𝐹‘𝑚)) → (𝑑 = {𝑎, 𝑏} ↔ 𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)}))
67	66	adantl 481	. . . . . . . . . . . . . 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 4692	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → {𝑥, 𝑦} = {𝑚, 𝑛})
69	68	eleq1d 2821	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑚, 𝑛} ∈ 𝐸))
70		fveq2 6834	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑚 → (𝐹‘𝑥) = (𝐹‘𝑚))
71	70	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → (𝐹‘𝑥) = (𝐹‘𝑚))
72		fveq2 6834	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑛 → (𝐹‘𝑦) = (𝐹‘𝑛))
73	72	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → (𝐹‘𝑦) = (𝐹‘𝑛))
74	71, 73	preq12d 4698	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → {(𝐹‘𝑥), (𝐹‘𝑦)} = {(𝐹‘𝑚), (𝐹‘𝑛)})
75	74	eleq1d 2821	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → ({(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷))
76	69, 75	bibi12d 345	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → (({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷) ↔ ({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷)))
77	76	rspc2gv 3586	. . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) → 𝐹 Fn 𝑉)
80	79	anim1i 615	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
81		3anass 1094	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ↔ (𝐹 Fn 𝑉 ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
82	80, 81	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ 𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉))
83		fnimapr 6917	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → (𝐹 “ {𝑚, 𝑛}) = {(𝐹‘𝑚), (𝐹‘𝑛)})
84	82, 83	syl 17	. . . . . . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ {𝑚, 𝑛}))
86		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 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		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → {𝑚, 𝑛} ∈ 𝐸)
88		reueq 3695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ({𝑚, 𝑛} ∈ 𝐸 ↔ ∃!𝑒 ∈ 𝐸 𝑒 = {𝑚, 𝑛})
89	87, 88	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → ∃!𝑒 ∈ 𝐸 𝑒 = {𝑚, 𝑛})
90		eqcom 2743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ({𝑚, 𝑛} = 𝑒 ↔ 𝑒 = {𝑚, 𝑛})
91	90	reubii 3359	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∃!𝑒 ∈ 𝐸 {𝑚, 𝑛} = 𝑒 ↔ ∃!𝑒 ∈ 𝐸 𝑒 = {𝑚, 𝑛})
92	89, 91	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → ∃!𝑒 ∈ 𝐸 {𝑚, 𝑛} = 𝑒)
93		f1of1 6773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹:𝑉–1-1→𝑊)
94	93	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) → 𝐹:𝑉–1-1→𝑊)
95	94	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
96		prssi 4777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → {𝑚, 𝑛} ⊆ 𝑉)
97	96	ad3antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) ∧ 𝑒 ∈ 𝐸) → {𝑚, 𝑛} ⊆ 𝑉)
98		uspgruhgr 29257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
99	98	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐺 ∈ UHGraph)
100	99	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → 𝐺 ∈ UHGraph)
101	6	eleq2i 2828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
102	101	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑒 ∈ 𝐸 → 𝑒 ∈ (Edg‘𝐺))
103		edguhgr 29202	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
104	5	pweqi 4570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
105	103, 104	eleqtrrdi 2847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 𝑉)
106	100, 102, 105	syl2an 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝒫 𝑉)
107	106	elpwid 4563	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) ∧ 𝑒 ∈ 𝐸) → 𝑒 ⊆ 𝑉)
108		f1imaeq 7211	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐹:𝑉–1-1→𝑊 ∧ ({𝑚, 𝑛} ⊆ 𝑉 ∧ 𝑒 ⊆ 𝑉)) → ((𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒) ↔ {𝑚, 𝑛} = 𝑒))
109	95, 97, 107, 108	syl12anc 836	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
110	109	reubidva 3364	. . . . . . . . . . . . . . . . . . . . . . . . . . 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))
111	92, 110	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . 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))
112	111	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . 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)))
113	112	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 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)))
114	86, 113	sylbird 260	. . . . . . . . . . . . . . . . . . . . . . 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)))
115	114	ex 412	. . . . . . . . . . . . . . . . . . . . . 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))))
116		eleq1 2824	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ {𝑚, 𝑛}) → ({(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷))
117	116	bibi2d 342	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ {𝑚, 𝑛}) → (({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷) ↔ ({𝑚, 𝑛} ∈ 𝐸 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷)))
118		eqeq1 2740	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ({(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ {𝑚, 𝑛}) → ({(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒) ↔ (𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒)))
119	118	reubidv 3366	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ {𝑚, 𝑛}) → (∃!𝑒 ∈ 𝐸 {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒) ↔ ∃!𝑒 ∈ 𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒)))
120	116, 119	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . 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))))
121	117, 120	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . 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)))))
122	115, 121	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . 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)))))
123	85, 122	mpd 15	. . . . . . . . . . . . . . . . . . . 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))))
124	78, 123	syld 47	. . . . . . . . . . . . . . . . . . 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))))
125	124	impancom 451	. . . . . . . . . . . . . . . . . 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))))
126	125	adantr 480	. . . . . . . . . . . . . . . . 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))))
127	126	impl 455	. . . . . . . . . . . . . . . 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)))
128		eleq1 2824	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)} → (𝑑 ∈ 𝐷 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷))
129		eqeq1 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)} → (𝑑 = (𝐹 “ 𝑒) ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒)))
130	129	reubidv 3366	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)} → (∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒) ↔ ∃!𝑒 ∈ 𝐸 {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒)))
131	128, 130	imbi12d 344	. . . . . . . . . . . . . . . 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))))
132	127, 131	syl5ibrcom 247	. . . . . . . . . . . . . . 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))))
133	132	adantr 480	. . . . . . . . . . . . . 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))))
134	67, 133	sylbid 240	. . . . . . . . . . . . 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))))
135	134	exp32 420	. . . . . . . . . . . 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))))))
136	135	rexlimdva 3137	. . . . . . . . . . 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))))))
137	136	com23 86	. . . . . . . . . 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))))))
138	137	rexlimdva 3137	. . . . . . . . 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))))))
139	138	impd 410	. . . . . . . 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)))))
140	63, 139	mpd 15	. . . . . . 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))))
141	140	com23 86	. . . . . 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))))
142	141	impancom 451	. . . . 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))))
143	142	rexlimdvv 3192	. . . 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)))
144	53, 143	mpd 15	. . 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))
145	144	ralrimiva 3128	. 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))
146		eqid 2736	. . 3 ⊢ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))
147	146	f1ompt 7056	. 2 ⊢ ((𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷 ↔ (∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) ∈ 𝐷 ∧ ∀𝑑 ∈ 𝐷 ∃!𝑒 ∈ 𝐸 𝑑 = (𝐹 “ 𝑒)))
148	47, 145, 147	sylanbrc 583	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  ∃!wreu 3348   ⊆ wss 3901  𝒫 cpw 4554  {cpr 4582   ↦ cmpt 5179   “ cima 5627   Fn wfn 6487  –1-1→wf1 6489  –onto→wfo 6490  –1-1-onto→wf1o 6491  ‘cfv 6492  Vtxcvtx 29069  Edgcedg 29120  UHGraphcuhgr 29129  UPGraphcupgr 29153  USPGraphcuspgr 29221

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  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  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-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  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-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9813  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-n0 12402  df-xnn0 12475  df-z 12489  df-uz 12752  df-fz 13424  df-hash 14254  df-edg 29121  df-uhgr 29131  df-upgr 29155  df-uspgr 29223

This theorem is referenced by:  isuspgrim  48138