Theorem edglnl 29721

Description: The edges incident with a vertex 𝑁 are the edges joining 𝑁 with other vertices and the loops on 𝑁 in a pseudograph. (Contributed by AV, 18-Dec-2021.)

Hypotheses

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

Assertion

Ref	Expression
edglnl	⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}) = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)})

Distinct variable groups:   𝑣,𝐸   𝑖,𝐺   𝑖,𝑁,𝑣   𝑖,𝑉,𝑣

Allowed substitution hints:   𝐸(𝑖)   𝐺(𝑣)

Proof of Theorem edglnl

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iunrab 5011	. . . 4 ⊢ ∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} = {𝑖 ∈ dom 𝐸 ∣ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))}
2	1	a1i 11	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} = {𝑖 ∈ dom 𝐸 ∣ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))})
3	2	uneq1d 4114	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}) = ({𝑖 ∈ dom 𝐸 ∣ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))
4		unrab 4261	. . 3 ⊢ ({𝑖 ∈ dom 𝐸 ∣ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}) = {𝑖 ∈ dom 𝐸 ∣ (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∨ (𝐸‘𝑖) = {𝑁})}
5		simpl 488	. . . . . . . 8 ⊢ ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) → 𝑁 ∈ (𝐸‘𝑖))
6	5	rexlimivw 3160	. . . . . . 7 ⊢ (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) → 𝑁 ∈ (𝐸‘𝑖))
7	6	a1i 11	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) → 𝑁 ∈ (𝐸‘𝑖)))
8		snidg 4621	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ {𝑁})
9	8	ad2antlr 740	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → 𝑁 ∈ {𝑁})
10		eleq2 2850	. . . . . . 7 ⊢ ((𝐸‘𝑖) = {𝑁} → (𝑁 ∈ (𝐸‘𝑖) ↔ 𝑁 ∈ {𝑁}))
11	9, 10	syl5ibrcom 250	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝐸‘𝑖) = {𝑁} → 𝑁 ∈ (𝐸‘𝑖)))
12	7, 11	jaod 873	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∨ (𝐸‘𝑖) = {𝑁}) → 𝑁 ∈ (𝐸‘𝑖)))
13		upgruhgr 29680	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
14		edglnl.e	. . . . . . . . . 10 ⊢ 𝐸 = (iEdg‘𝐺)
15	14	uhgrfun 29644	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
16	13, 15	syl 18	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → Fun 𝐸)
17	16	adantr 486	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → Fun 𝐸)
18	14	iedgedg 29628	. . . . . . 7 ⊢ ((Fun 𝐸 ∧ 𝑖 ∈ dom 𝐸) → (𝐸‘𝑖) ∈ (Edg‘𝐺))
19	17, 18	sylan 592	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → (𝐸‘𝑖) ∈ (Edg‘𝐺))
20		edglnl.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
21		eqid 2761	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
22	20, 21	upgredg 29715	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐸‘𝑖) ∈ (Edg‘𝐺)) → ∃𝑛 ∈ 𝑉 ∃𝑚 ∈ 𝑉 (𝐸‘𝑖) = {𝑛, 𝑚})
23	22	ex 418	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → ((𝐸‘𝑖) ∈ (Edg‘𝐺) → ∃𝑛 ∈ 𝑉 ∃𝑚 ∈ 𝑉 (𝐸‘𝑖) = {𝑛, 𝑚}))
24	23	ad2antrr 739	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝐸‘𝑖) ∈ (Edg‘𝐺) → ∃𝑛 ∈ 𝑉 ∃𝑚 ∈ 𝑉 (𝐸‘𝑖) = {𝑛, 𝑚}))
25		dfsn2 4597	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {𝑛} = {𝑛, 𝑛}
26	25	eqcomi 2770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑛, 𝑛} = {𝑛}
27		elsni 4601	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ {𝑛} → 𝑁 = 𝑛)
28		sneq 4594	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 = 𝑛 → {𝑁} = {𝑛})
29	28	eqcomd 2767	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 = 𝑛 → {𝑛} = {𝑁})
30	27, 29	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ {𝑛} → {𝑛} = {𝑁})
31	26, 30	eqtrid 2808	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ {𝑛} → {𝑛, 𝑛} = {𝑁})
32	31, 26	eleq2s 2879	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ {𝑛, 𝑛} → {𝑛, 𝑛} = {𝑁})
33		preq2 4695	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → {𝑛, 𝑚} = {𝑛, 𝑛})
34	33	eleq2d 2847	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → (𝑁 ∈ {𝑛, 𝑚} ↔ 𝑁 ∈ {𝑛, 𝑛}))
35	33	eqeq1d 2763	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → ({𝑛, 𝑚} = {𝑁} ↔ {𝑛, 𝑛} = {𝑁}))
36	34, 35	imbi12d 347	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑛 → ((𝑁 ∈ {𝑛, 𝑚} → {𝑛, 𝑚} = {𝑁}) ↔ (𝑁 ∈ {𝑛, 𝑛} → {𝑛, 𝑛} = {𝑁})))
37	32, 36	mpbiri 261	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → (𝑁 ∈ {𝑛, 𝑚} → {𝑛, 𝑚} = {𝑁}))
38	37	imp 412	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 = 𝑛 ∧ 𝑁 ∈ {𝑛, 𝑚}) → {𝑛, 𝑚} = {𝑁})
39	38	olcd 888	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 = 𝑛 ∧ 𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))
40	39	expcom 419	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ {𝑛, 𝑚} → (𝑚 = 𝑛 → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
41	40	3ad2ant3 1153	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚}) → (𝑚 = 𝑛 → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
42	41	com12 33	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
43		simpr3 1215	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑁 ∈ {𝑛, 𝑚})
44		simpl 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑚 ≠ 𝑛)
45	44	necomd 3011	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑛 ≠ 𝑚)
46		simpr2 1214	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉))
47		prproe 4865	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ {𝑛, 𝑚} ∧ 𝑛 ≠ 𝑚 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚})
48	43, 45, 46, 47	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚})
49		r19.42v 3195	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ↔ (𝑁 ∈ {𝑛, 𝑚} ∧ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚}))
50	43, 48, 49	sylanbrc 595	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}))
51	50	orcd 887	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))
52	51	ex 418	. . . . . . . . . . . . 13 ⊢ (𝑚 ≠ 𝑛 → ((𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
53	42, 52	pm2.61ine 3039	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))
54	53	3exp 1137	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑉 → ((𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) → (𝑁 ∈ {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))))
55	54	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) → (𝑁 ∈ {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))))
56	55	imp 412	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉)) → (𝑁 ∈ {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
57		eleq2 2850	. . . . . . . . . 10 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → (𝑁 ∈ (𝐸‘𝑖) ↔ 𝑁 ∈ {𝑛, 𝑚}))
58		eleq2 2850	. . . . . . . . . . . . 13 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → (𝑣 ∈ (𝐸‘𝑖) ↔ 𝑣 ∈ {𝑛, 𝑚}))
59	57, 58	anbi12d 644	. . . . . . . . . . . 12 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ↔ (𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚})))
60	59	rexbidv 3187	. . . . . . . . . . 11 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ↔ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚})))
61		eqeq1 2765	. . . . . . . . . . 11 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → ((𝐸‘𝑖) = {𝑁} ↔ {𝑛, 𝑚} = {𝑁}))
62	60, 61	orbi12d 932	. . . . . . . . . 10 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → ((∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∨ (𝐸‘𝑖) = {𝑁}) ↔ (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
63	57, 62	imbi12d 347	. . . . . . . . 9 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → ((𝑁 ∈ (𝐸‘𝑖) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∨ (𝐸‘𝑖) = {𝑁})) ↔ (𝑁 ∈ {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))))
64	56, 63	syl5ibrcom 250	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉)) → ((𝐸‘𝑖) = {𝑛, 𝑚} → (𝑁 ∈ (𝐸‘𝑖) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∨ (𝐸‘𝑖) = {𝑁}))))
65	64	rexlimdvva 3220	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → (∃𝑛 ∈ 𝑉 ∃𝑚 ∈ 𝑉 (𝐸‘𝑖) = {𝑛, 𝑚} → (𝑁 ∈ (𝐸‘𝑖) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∨ (𝐸‘𝑖) = {𝑁}))))
66	24, 65	syld 48	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝐸‘𝑖) ∈ (Edg‘𝐺) → (𝑁 ∈ (𝐸‘𝑖) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∨ (𝐸‘𝑖) = {𝑁}))))
67	19, 66	mpd 16	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → (𝑁 ∈ (𝐸‘𝑖) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∨ (𝐸‘𝑖) = {𝑁})))
68	12, 67	impbid 215	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∨ (𝐸‘𝑖) = {𝑁}) ↔ 𝑁 ∈ (𝐸‘𝑖)))
69	68	rabbidva 3419	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → {𝑖 ∈ dom 𝐸 ∣ (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∨ (𝐸‘𝑖) = {𝑁})} = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)})
70	4, 69	eqtrid 2808	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ({𝑖 ∈ dom 𝐸 ∣ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}) = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)})
71	3, 70	eqtrd 2796	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}) = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087  {crab 3413   ∖ cdif 3896   ∪ cun 3897  {csn 4584  {cpr 4586  ∪ ciun 4951  dom cdm 5651  Fun wfun 6532  ‘cfv 6538  Vtxcvtx 29574  iEdgciedg 29575  Edgcedg 29625  UHGraphcuhgr 29634  UPGraphcupgr 29658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-dju 9982  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-n0 12607  df-xnn0 12680  df-z 12694  df-uz 12966  df-fz 13640  df-hash 14475  df-edg 29626  df-uhgr 29636  df-upgr 29660

This theorem is used by:  numedglnl  29722