Theorem upgrex 29071

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 29068	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ≠ ∅)
4		n0 4303	. . . 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 1136	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐺 ∈ UPGraph)
7		fndm 6584	. . . . . . . . . . . . 13 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
8	7	eqcomd 2737	. . . . . . . . . . . 12 ⊢ (𝐸 Fn 𝐴 → 𝐴 = dom 𝐸)
9	8	eleq2d 2817	. . . . . . . . . . 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 1110	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐹 ∈ dom 𝐸)
13	1, 2	upgrss 29067	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉)
14	6, 12, 13	syl2anc 584	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ⊆ 𝑉)
15	14	sselda 3934	. . . . . 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 4316	. . . . . . . . . 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 4761	. . . . . . . . . 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 3952	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ((𝐸‘𝐹) ∖ {𝑥}) = ∅) → (𝐸‘𝐹) = {𝑥})
24		preq2 4687	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → {𝑥, 𝑦} = {𝑥, 𝑥})
25		dfsn2 4589	. . . . . . . . . 10 ⊢ {𝑥} = {𝑥, 𝑥}
26	24, 25	eqtr4di 2784	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → {𝑥, 𝑦} = {𝑥})
27	26	rspceeqv 3600	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝐸‘𝐹) = {𝑥}) → ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})
28	16, 23, 27	syl2anc 584	. . . . . . 7 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ((𝐸‘𝐹) ∖ {𝑥}) = ∅) → ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})
29		n0 4303	. . . . . . . 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 772	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))
32	31	eldifad 3914	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑦 ∈ (𝐸‘𝐹))
33	30, 32	sseldd 3935	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑦 ∈ 𝑉)
34	1, 2	upgrfi 29070	. . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑥 ∈ (𝐸‘𝐹))
37	36, 32	prssd 4774	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} ⊆ (𝐸‘𝐹))
38		fvex 6835	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸‘𝐹) ∈ V
39		ssdomg 8922	. . . . . . . . . . . . . . . . 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 29069	. . . . . . . . . . . . . . . . . . 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 4742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}) → 𝑦 ≠ 𝑥)
44	43	ad2antll 729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑦 ≠ 𝑥)
45	44	necomd 2983	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑥 ≠ 𝑦)
46		hashprg 14302	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≠ 𝑦 ↔ (♯‘{𝑥, 𝑦}) = 2))
47	46	el2v 3443	. . . . . . . . . . . . . . . . . . 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 5119	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (♯‘(𝐸‘𝐹)) ≤ (♯‘{𝑥, 𝑦}))
50		prfi 9208	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑥, 𝑦} ∈ Fin
51		hashdom 14286	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐸‘𝐹) ∈ Fin ∧ {𝑥, 𝑦} ∈ Fin) → ((♯‘(𝐸‘𝐹)) ≤ (♯‘{𝑥, 𝑦}) ↔ (𝐸‘𝐹) ≼ {𝑥, 𝑦}))
52	35, 50, 51	sylancl 586	. . . . . . . . . . . . . . . . 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 9010	. . . . . . . . . . . . . . . 16 ⊢ (({𝑥, 𝑦} ≼ (𝐸‘𝐹) ∧ (𝐸‘𝐹) ≼ {𝑥, 𝑦}) → {𝑥, 𝑦} ≈ (𝐸‘𝐹))
55	40, 53, 54	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} ≈ (𝐸‘𝐹))
56		fisseneq 9147	. . . . . . . . . . . . . . 15 ⊢ (((𝐸‘𝐹) ∈ Fin ∧ {𝑥, 𝑦} ⊆ (𝐸‘𝐹) ∧ {𝑥, 𝑦} ≈ (𝐸‘𝐹)) → {𝑥, 𝑦} = (𝐸‘𝐹))
57	35, 37, 55, 56	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} = (𝐸‘𝐹))
58	57	eqcomd 2737	. . . . . . . . . . . . 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 1918	. . . . . . . . . 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 3057	. . . . . . . . 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 594	. . . . . . 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 3015	. . . . . 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 1918	. . 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 3057	. 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 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056  Vcvv 3436   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4283  {csn 4576  {cpr 4578   class class class wbr 5091  dom cdm 5616   Fn wfn 6476  ‘cfv 6481   ≈ cen 8866   ≼ cdom 8867  Fincfn 8869   ≤ cle 11147  2c2 12180  ♯chash 14237  Vtxcvtx 28975  iEdgciedg 28976  UPGraphcupgr 29059

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-oadd 8389  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-dju 9794  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-n0 12382  df-xnn0 12455  df-z 12469  df-uz 12733  df-fz 13408  df-hash 14238  df-upgr 29061

This theorem is referenced by: (None)