Theorem upgrex 28617

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 28614	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ≠ ∅)
4		n0 4347	. . . 4 ⊢ ((𝐸‘𝐹) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐸‘𝐹))
5	3, 4	sylib 217	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ∃𝑥 𝑥 ∈ (𝐸‘𝐹))
6		simp1 1134	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐺 ∈ UPGraph)
7		fndm 6653	. . . . . . . . . . . . 13 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
8	7	eqcomd 2736	. . . . . . . . . . . 12 ⊢ (𝐸 Fn 𝐴 → 𝐴 = dom 𝐸)
9	8	eleq2d 2817	. . . . . . . . . . 11 ⊢ (𝐸 Fn 𝐴 → (𝐹 ∈ 𝐴 ↔ 𝐹 ∈ dom 𝐸))
10	9	biimpd 228	. . . . . . . . . 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 1109	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐹 ∈ dom 𝐸)
13	1, 2	upgrss 28613	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉)
14	6, 12, 13	syl2anc 582	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ⊆ 𝑉)
15	14	sselda 3983	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) → 𝑥 ∈ 𝑉)
16	15	adantr 479	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ((𝐸‘𝐹) ∖ {𝑥}) = ∅) → 𝑥 ∈ 𝑉)
17		simpr 483	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ((𝐸‘𝐹) ∖ {𝑥}) = ∅) → ((𝐸‘𝐹) ∖ {𝑥}) = ∅)
18		ssdif0 4364	. . . . . . . . . 10 ⊢ ((𝐸‘𝐹) ⊆ {𝑥} ↔ ((𝐸‘𝐹) ∖ {𝑥}) = ∅)
19	17, 18	sylibr 233	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ((𝐸‘𝐹) ∖ {𝑥}) = ∅) → (𝐸‘𝐹) ⊆ {𝑥})
20		simpr 483	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) → 𝑥 ∈ (𝐸‘𝐹))
21	20	snssd 4813	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) → {𝑥} ⊆ (𝐸‘𝐹))
22	21	adantr 479	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ((𝐸‘𝐹) ∖ {𝑥}) = ∅) → {𝑥} ⊆ (𝐸‘𝐹))
23	19, 22	eqssd 4000	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ((𝐸‘𝐹) ∖ {𝑥}) = ∅) → (𝐸‘𝐹) = {𝑥})
24		preq2 4739	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → {𝑥, 𝑦} = {𝑥, 𝑥})
25		dfsn2 4642	. . . . . . . . . 10 ⊢ {𝑥} = {𝑥, 𝑥}
26	24, 25	eqtr4di 2788	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → {𝑥, 𝑦} = {𝑥})
27	26	rspceeqv 3634	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ (𝐸‘𝐹) = {𝑥}) → ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})
28	16, 23, 27	syl2anc 582	. . . . . . 7 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) ∧ ((𝐸‘𝐹) ∖ {𝑥}) = ∅) → ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})
29		n0 4347	. . . . . . . 8 ⊢ (((𝐸‘𝐹) ∖ {𝑥}) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))
30	14	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (𝐸‘𝐹) ⊆ 𝑉)
31		simprr 769	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))
32	31	eldifad 3961	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑦 ∈ (𝐸‘𝐹))
33	30, 32	sseldd 3984	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑦 ∈ 𝑉)
34	1, 2	upgrfi 28616	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ Fin)
35	34	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (𝐸‘𝐹) ∈ Fin)
36		simprl 767	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑥 ∈ (𝐸‘𝐹))
37	36, 32	prssd 4826	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} ⊆ (𝐸‘𝐹))
38		fvex 6905	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸‘𝐹) ∈ V
39		ssdomg 9000	. . . . . . . . . . . . . . . . 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 28615	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (♯‘(𝐸‘𝐹)) ≤ 2)
42	41	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (♯‘(𝐸‘𝐹)) ≤ 2)
43		eldifsni 4794	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}) → 𝑦 ≠ 𝑥)
44	43	ad2antll 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑦 ≠ 𝑥)
45	44	necomd 2994	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → 𝑥 ≠ 𝑦)
46		hashprg 14361	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≠ 𝑦 ↔ (♯‘{𝑥, 𝑦}) = 2))
47	46	el2v 3480	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ≠ 𝑦 ↔ (♯‘{𝑥, 𝑦}) = 2)
48	45, 47	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (♯‘{𝑥, 𝑦}) = 2)
49	42, 48	breqtrrd 5177	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (♯‘(𝐸‘𝐹)) ≤ (♯‘{𝑥, 𝑦}))
50		prfi 9326	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑥, 𝑦} ∈ Fin
51		hashdom 14345	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐸‘𝐹) ∈ Fin ∧ {𝑥, 𝑦} ∈ Fin) → ((♯‘(𝐸‘𝐹)) ≤ (♯‘{𝑥, 𝑦}) ↔ (𝐸‘𝐹) ≼ {𝑥, 𝑦}))
52	35, 50, 51	sylancl 584	. . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (𝐸‘𝐹) ≼ {𝑥, 𝑦})
54		sbth 9097	. . . . . . . . . . . . . . . 16 ⊢ (({𝑥, 𝑦} ≼ (𝐸‘𝐹) ∧ (𝐸‘𝐹) ≼ {𝑥, 𝑦}) → {𝑥, 𝑦} ≈ (𝐸‘𝐹))
55	40, 53, 54	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} ≈ (𝐸‘𝐹))
56		fisseneq 9261	. . . . . . . . . . . . . . 15 ⊢ (((𝐸‘𝐹) ∈ Fin ∧ {𝑥, 𝑦} ⊆ (𝐸‘𝐹) ∧ {𝑥, 𝑦} ≈ (𝐸‘𝐹)) → {𝑥, 𝑦} = (𝐸‘𝐹))
57	35, 37, 55, 56	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} = (𝐸‘𝐹))
58	57	eqcomd 2736	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (𝐸‘𝐹) = {𝑥, 𝑦})
59	33, 58	jca 510	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝑥 ∈ (𝐸‘𝐹) ∧ 𝑦 ∈ ((𝐸‘𝐹) ∖ {𝑥}))) → (𝑦 ∈ 𝑉 ∧ (𝐸‘𝐹) = {𝑥, 𝑦}))
60	59	expr 455	. . . . . . . . . . 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 405	. . . . . . . . 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 3069	. . . . . . . . 9 ⊢ (∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦} ↔ ∃𝑦(𝑦 ∈ 𝑉 ∧ (𝐸‘𝐹) = {𝑥, 𝑦}))
64	62, 63	sylibr 233	. . . . . . . 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 592	. . . . . . 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 3027	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) → ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})
67	15, 66	jca 510	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ 𝑥 ∈ (𝐸‘𝐹)) → (𝑥 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦}))
68	67	ex 411	. . . 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 3069	. 2 ⊢ (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦} ↔ ∃𝑥(𝑥 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦}))
72	70, 71	sylibr 233	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539  ∃wex 1779   ∈ wcel 2104   ≠ wne 2938  ∃wrex 3068  Vcvv 3472   ∖ cdif 3946   ⊆ wss 3949  ∅c0 4323  {csn 4629  {cpr 4631   class class class wbr 5149  dom cdm 5677   Fn wfn 6539  ‘cfv 6544   ≈ cen 8940   ≼ cdom 8941  Fincfn 8943   ≤ cle 11255  2c2 12273  ♯chash 14296  Vtxcvtx 28521  iEdgciedg 28522  UPGraphcupgr 28605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-oadd 8474  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9900  df-card 9938  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11452  df-neg 11453  df-nn 12219  df-2 12281  df-n0 12479  df-xnn0 12551  df-z 12565  df-uz 12829  df-fz 13491  df-hash 14297  df-upgr 28607

This theorem is referenced by: (None)