Theorem isuspgrimlem 48249

Description: Lemma for isuspgrim 48250. (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 29263	. . . . . . . . 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 29222	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑒 = {𝑎, 𝑏})
8	4, 7	sylan 581	. . . . 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 4694	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
10	9	eleq1d 2822	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
11		fveq2 6842	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
12	11	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝐹‘𝑥) = (𝐹‘𝑎))
13		fveq2 6842	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑏 → (𝐹‘𝑦) = (𝐹‘𝑏))
14	13	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝐹‘𝑦) = (𝐹‘𝑏))
15	12, 14	preq12d 4700	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → {(𝐹‘𝑥), (𝐹‘𝑦)} = {(𝐹‘𝑎), (𝐹‘𝑏)})
16	15	eleq1d 2822	. . . . . . . . . . . . . . . . . . 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 3588	. . . . . . . . . . . . . . . . 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 6783	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹 Fn 𝑉)
23	22	ad3antlr 732	. . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . 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 6925	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐹 “ {𝑎, 𝑏}) = {(𝐹‘𝑎), (𝐹‘𝑏)})
28	23, 24, 26, 27	syl3anc 1374	. . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . 13 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑒 ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
35		imaeq2 6023	. . . . . . . . . . . . . 14 ⊢ (𝑒 = {𝑎, 𝑏} → (𝐹 “ 𝑒) = (𝐹 “ {𝑎, 𝑏}))
36	35	eleq1d 2822	. . . . . . . . . . . . 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 3194	. . . . 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 3129	. 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 29263	. . . . . 6 ⊢ (𝐻 ∈ USPGraph → 𝐻 ∈ UPGraph)
49	48	ad3antlr 732	. . . . 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 29222	. . . . 5 ⊢ ((𝐻 ∈ UPGraph ∧ 𝑑 ∈ 𝐷) → ∃𝑎 ∈ 𝑊 ∃𝑏 ∈ 𝑊 𝑑 = {𝑎, 𝑏})
53	49, 52	sylan 581	. . . 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 6789	. . . . . . . . . . . 12 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹:𝑉–onto→𝑊)
55		foelrn 7061	. . . . . . . . . . . . . 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 7061	. . . . . . . . . . . . . 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 610	. . . . . . . . . . . 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 4694	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = (𝐹‘𝑚) ∧ 𝑏 = (𝐹‘𝑛)) → {𝑎, 𝑏} = {(𝐹‘𝑚), (𝐹‘𝑛)})
65	64	eqeq2d 2748	. . . . . . . . . . . . . . . 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 4694	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → {𝑥, 𝑦} = {𝑚, 𝑛})
69	68	eleq1d 2822	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑚, 𝑛} ∈ 𝐸))
70		fveq2 6842	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑚 → (𝐹‘𝑥) = (𝐹‘𝑚))
71	70	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → (𝐹‘𝑥) = (𝐹‘𝑚))
72		fveq2 6842	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑛 → (𝐹‘𝑦) = (𝐹‘𝑛))
73	72	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → (𝐹‘𝑦) = (𝐹‘𝑛))
74	71, 73	preq12d 4700	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 = 𝑚 ∧ 𝑦 = 𝑛) → {(𝐹‘𝑥), (𝐹‘𝑦)} = {(𝐹‘𝑚), (𝐹‘𝑛)})
75	74	eleq1d 2822	. . . . . . . . . . . . . . . . . . . . . . 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 3588	. . . . . . . . . . . . . . . . . . . . 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 616	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
81		3anass 1095	. . . . . . . . . . . . . . . . . . . . . . . 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 6925	. . . . . . . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . . . . 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 3697	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ({𝑚, 𝑛} = 𝑒 ↔ 𝑒 = {𝑚, 𝑛})
91	90	reubii 3361	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6781	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → {𝑚, 𝑛} ⊆ 𝑉)
97	96	ad3antlr 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 29269	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
99	98	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐺 ∈ UHGraph)
100	99	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → 𝐺 ∈ UHGraph)
101	6	eleq2i 2829	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
102	101	biimpi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑒 ∈ 𝐸 → 𝑒 ∈ (Edg‘𝐺))
103		edguhgr 29214	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
104	5	pweqi 4572	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
105	103, 104	eleqtrrdi 2848	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 𝑉)
106	100, 102, 105	syl2an 597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4565	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) ∧ 𝑒 ∈ 𝐸) → 𝑒 ⊆ 𝑉)
108		f1imaeq 7221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3366	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ({(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ {𝑚, 𝑛}) → ({(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒) ↔ (𝐹 “ {𝑚, 𝑛}) = (𝐹 “ 𝑒)))
119	118	reubidv 3368	. . . . . . . . . . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)} → (𝑑 ∈ 𝐷 ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} ∈ 𝐷))
129		eqeq1 2741	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = {(𝐹‘𝑚), (𝐹‘𝑛)} → (𝑑 = (𝐹 “ 𝑒) ↔ {(𝐹‘𝑚), (𝐹‘𝑛)} = (𝐹 “ 𝑒)))
130	129	reubidv 3368	. . . . . . . . . . . . . . . . 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 3139	. . . . . . . . . . 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 3139	. . . . . . . . 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 3194	. . . 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 3130	. 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 2737	. . 3 ⊢ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)) = (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒))
147	146	f1ompt 7065	. 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 584	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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  ∃!wreu 3350   ⊆ wss 3903  𝒫 cpw 4556  {cpr 4584   ↦ cmpt 5181   “ cima 5635   Fn wfn 6495  –1-1→wf1 6497  –onto→wfo 6498  –1-1-onto→wf1o 6499  ‘cfv 6500  Vtxcvtx 29081  Edgcedg 29132  UHGraphcuhgr 29141  UPGraphcupgr 29165  USPGraphcuspgr 29233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-n0 12414  df-xnn0 12487  df-z 12501  df-uz 12764  df-fz 13436  df-hash 14266  df-edg 29133  df-uhgr 29143  df-upgr 29167  df-uspgr 29235

This theorem is referenced by:  isuspgrim  48250