Theorem upgrex 29161

Description: An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)

Hypotheses

Ref	Expression
isupgr.v	⊢ 𝑉 = (Vtx‘𝐺)
isupgr.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
upgrex	⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})

Distinct variable groups:   𝑥,𝐺   𝑥,𝑉   𝑥,𝐸   𝑥,𝐹   𝑥,𝐴,𝑦   𝑦,𝐸   𝑦,𝐹   𝑦,𝐺   𝑦,𝑉

Proof of Theorem upgrex

Step	Hyp	Ref	Expression
1		isupgr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		isupgr.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	upgrn0 29158	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ≠ ∅)
4		n0 4293	. . . 4 ⊢ ((𝐸‘𝐹) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐸‘𝐹))
5	3, 4	sylib 218	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ∃𝑥 𝑥 ∈ (𝐸‘𝐹))
6		simp1 1137	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐺 ∈ UPGraph)
7		fndm 6601	. . . . . . . . . . . . 13 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
8	7	eqcomd 2742	. . . . . . . . . . . 12 ⊢ (𝐸 Fn 𝐴 → 𝐴 = dom 𝐸)
9	8	eleq2d 2822	. . . . . . . . . . 11 ⊢ (𝐸 Fn 𝐴 → (𝐹 ∈ 𝐴 ↔ 𝐹 ∈ dom 𝐸))
10	9	biimpd 229	. . . . . . . . . 10 ⊢ (𝐸 Fn 𝐴 → (𝐹 ∈ 𝐴 → 𝐹 ∈ dom 𝐸))
11	10	a1i 11	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → (𝐸 Fn 𝐴 → (𝐹 ∈ 𝐴 → 𝐹 ∈ dom 𝐸)))
12	11	3imp 1111	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐹 ∈ dom 𝐸)
13	1, 2	upgrss 29157	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉)
14	6, 12, 13	syl2anc 585	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ⊆ 𝑉)
15	14	sselda 3921	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) → 𝑥 ∈ 𝑉)
16	15	adantr 480	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ((𝐸‘𝐹) ∖ {𝑥}) = ∅) → 𝑥 ∈ 𝑉)
17		simpr 484	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ((𝐸‘𝐹) ∖ {𝑥}) = ∅) → ((𝐸‘𝐹) ∖ {𝑥}) = ∅)
18		ssdif0 4306	. . . . . . . . . 10 ⊢ ((𝐸‘𝐹) ⊆ {𝑥} ↔ ((𝐸‘𝐹) ∖ {𝑥}) = ∅)
19	17, 18	sylibr 234	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ((𝐸‘𝐹) ∖ {𝑥}) = ∅) → (𝐸‘𝐹) ⊆ {𝑥})
20		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) → 𝑥 ∈ (𝐸‘𝐹))
21	20	snssd 4730	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) → {𝑥} ⊆ (𝐸‘𝐹))
22	21	adantr 480	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ((𝐸‘𝐹) ∖ {𝑥}) = ∅) → {𝑥} ⊆ (𝐸‘𝐹))
23	19, 22	eqssd 3939	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ((𝐸‘𝐹) ∖ {𝑥}) = ∅) → (𝐸‘𝐹) = {𝑥})
24		preq2 4678	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → {𝑥, 𝑦} = {𝑥, 𝑥})
25		dfsn2 4580	. . . . . . . . . 10 ⊢ {𝑥} = {𝑥, 𝑥}
26	24, 25	eqtr4di 2789	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → {𝑥, 𝑦} = {𝑥})
27	26	rspceeqv 3587	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝐸‘𝐹) = {𝑥}) → ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})
28	16, 23, 27	syl2anc 585	. . . . . . 7 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ((𝐸‘𝐹) ∖ {𝑥}) = ∅) → ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})
29		n0 4293	. . . . . . . 8 ⊢ (((𝐸‘𝐹) ∖ {𝑥}) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))
30	14	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (𝐸‘𝐹) ⊆ 𝑉)
31		simprr 773	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))
32	31	eldifad 3901	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑦 ∈ (𝐸‘𝐹))
33	30, 32	sseldd 3922	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑦 ∈ 𝑉)
34	1, 2	upgrfi 29160	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ Fin)
35	34	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (𝐸‘𝐹) ∈ Fin)
36		simprl 771	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑥 ∈ (𝐸‘𝐹))
37	36, 32	prssd 4765	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} ⊆ (𝐸‘𝐹))
38		fvex 6853	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸‘𝐹) ∈ V
39		ssdomg 8947	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸‘𝐹) ∈ V → ({𝑥, 𝑦} ⊆ (𝐸‘𝐹) → {𝑥, 𝑦} ≼ (𝐸‘𝐹)))
40	38, 37, 39	mpsyl 68	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} ≼ (𝐸‘𝐹))
41	1, 2	upgrle 29159	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (♯‘(𝐸‘𝐹)) ≤ 2)
42	41	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (♯‘(𝐸‘𝐹)) ≤ 2)
43		eldifsni 4735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}) → 𝑦 ≠ 𝑥)
44	43	ad2antll 730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑦 ≠ 𝑥)
45	44	necomd 2987	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑥 ≠ 𝑦)
46		hashprg 14357	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≠ 𝑦 ↔ (♯‘{𝑥, 𝑦}) = 2))
47	46	el2v 3436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ≠ 𝑦 ↔ (♯‘{𝑥, 𝑦}) = 2)
48	45, 47	sylib 218	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (♯‘{𝑥, 𝑦}) = 2)
49	42, 48	breqtrrd 5113	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (♯‘(𝐸‘𝐹)) ≤ (♯‘{𝑥, 𝑦}))
50		prfi 9234	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑥, 𝑦} ∈ Fin
51		hashdom 14341	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐸‘𝐹) ∈ Fin ∧ {𝑥, 𝑦} ∈ Fin) → ((♯‘(𝐸‘𝐹)) ≤ (♯‘{𝑥, 𝑦}) ↔ (𝐸‘𝐹) ≼ {𝑥, 𝑦}))
52	35, 50, 51	sylancl 587	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → ((♯‘(𝐸‘𝐹)) ≤ (♯‘{𝑥, 𝑦}) ↔ (𝐸‘𝐹) ≼ {𝑥, 𝑦}))
53	49, 52	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (𝐸‘𝐹) ≼ {𝑥, 𝑦})
54		sbth 9035	. . . . . . . . . . . . . . . 16 ⊢ (({𝑥, 𝑦} ≼ (𝐸‘𝐹) ∧ (𝐸‘𝐹) ≼ {𝑥, 𝑦}) → {𝑥, 𝑦} ≈ (𝐸‘𝐹))
55	40, 53, 54	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} ≈ (𝐸‘𝐹))
56		fisseneq 9173	. . . . . . . . . . . . . . 15 ⊢ (((𝐸‘𝐹) ∈ Fin ∧ {𝑥, 𝑦} ⊆ (𝐸‘𝐹) ∧ {𝑥, 𝑦} ≈ (𝐸‘𝐹)) → {𝑥, 𝑦} = (𝐸‘𝐹))
57	35, 37, 55, 56	syl3anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} = (𝐸‘𝐹))
58	57	eqcomd 2742	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (𝐸‘𝐹) = {𝑥, 𝑦})
59	33, 58	jca 511	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (𝑦 ∈ 𝑉 ∧ (𝐸‘𝐹) = {𝑥, 𝑦}))
60	59	expr 456	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) → (𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}) → (𝑦 ∈ 𝑉 ∧ (𝐸‘𝐹) = {𝑥, 𝑦})))
61	60	eximdv 1919	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) → (∃𝑦 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}) → ∃𝑦(𝑦 ∈ 𝑉 ∧ (𝐸‘𝐹) = {𝑥, 𝑦})))
62	61	imp 406	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ∃𝑦 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥})) → ∃𝑦(𝑦 ∈ 𝑉 ∧ (𝐸‘𝐹) = {𝑥, 𝑦}))
63		df-rex 3062	. . . . . . . . 9 ⊢ (∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦} ↔ ∃𝑦(𝑦 ∈ 𝑉 ∧ (𝐸‘𝐹) = {𝑥, 𝑦}))
64	62, 63	sylibr 234	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ∃𝑦 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥})) → ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})
65	29, 64	sylan2b 595	. . . . . . 7 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ((𝐸‘𝐹) ∖ {𝑥}) ≠ ∅) → ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})
66	28, 65	pm2.61dane 3019	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) → ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})
67	15, 66	jca 511	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) → (𝑥 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦}))
68	67	ex 412	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝑥 ∈ (𝐸‘𝐹) → (𝑥 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})))
69	68	eximdv 1919	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (∃𝑥 𝑥 ∈ (𝐸‘𝐹) → ∃𝑥(𝑥 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})))
70	5, 69	mpd 15	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ∃𝑥(𝑥 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦}))
71		df-rex 3062	. 2 ⊢ (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦} ↔ ∃𝑥(𝑥 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦}))
72	70, 71	sylibr 234	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061  Vcvv 3429   ∖ cdif 3886   ⊆ wss 3889  ∅c0 4273  {csn 4567  {cpr 4569   class class class wbr 5085  dom cdm 5631   Fn wfn 6493  ‘cfv 6498   ≈ cen 8890   ≼ cdom 8891  Fincfn 8893   ≤ cle 11180  2c2 12236  ♯chash 14292  Vtxcvtx 29065  iEdgciedg 29066  UPGraphcupgr 29149

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 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  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 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 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-fz 13462  df-hash 14293  df-upgr 29151

This theorem is referenced by: (None)