Theorem edglnl 29088

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 5001	. . . 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 4118	. 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 4266	. . 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 482	. . . . . . . 8 ⊢ ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) → 𝑁 ∈ (𝐸‘𝑖))
6	5	rexlimivw 3126	. . . . . . 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 4612	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ {𝑁})
9	8	ad2antlr 727	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → 𝑁 ∈ {𝑁})
10		eleq2 2817	. . . . . . 7 ⊢ ((𝐸‘𝑖) = {𝑁} → (𝑁 ∈ (𝐸‘𝑖) ↔ 𝑁 ∈ {𝑁}))
11	9, 10	syl5ibrcom 247	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝐸‘𝑖) = {𝑁} → 𝑁 ∈ (𝐸‘𝑖)))
12	7, 11	jaod 859	. . . . 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 29047	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
14		edglnl.e	. . . . . . . . . 10 ⊢ 𝐸 = (iEdg‘𝐺)
15	14	uhgrfun 29011	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
16	13, 15	syl 17	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → Fun 𝐸)
17	16	adantr 480	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → Fun 𝐸)
18	14	iedgedg 28995	. . . . . . 7 ⊢ ((Fun 𝐸 ∧ 𝑖 ∈ dom 𝐸) → (𝐸‘𝑖) ∈ (Edg‘𝐺))
19	17, 18	sylan 580	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → (𝐸‘𝑖) ∈ (Edg‘𝐺))
20		edglnl.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
21		eqid 2729	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
22	20, 21	upgredg 29082	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐸‘𝑖) ∈ (Edg‘𝐺)) → ∃𝑛 ∈ 𝑉 ∃𝑚 ∈ 𝑉 (𝐸‘𝑖) = {𝑛, 𝑚})
23	22	ex 412	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → ((𝐸‘𝑖) ∈ (Edg‘𝐺) → ∃𝑛 ∈ 𝑉 ∃𝑚 ∈ 𝑉 (𝐸‘𝑖) = {𝑛, 𝑚}))
24	23	ad2antrr 726	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝐸‘𝑖) ∈ (Edg‘𝐺) → ∃𝑛 ∈ 𝑉 ∃𝑚 ∈ 𝑉 (𝐸‘𝑖) = {𝑛, 𝑚}))
25		dfsn2 4590	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {𝑛} = {𝑛, 𝑛}
26	25	eqcomi 2738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑛, 𝑛} = {𝑛}
27		elsni 4594	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ {𝑛} → 𝑁 = 𝑛)
28		sneq 4587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 = 𝑛 → {𝑁} = {𝑛})
29	28	eqcomd 2735	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 = 𝑛 → {𝑛} = {𝑁})
30	27, 29	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ {𝑛} → {𝑛} = {𝑁})
31	26, 30	eqtrid 2776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ {𝑛} → {𝑛, 𝑛} = {𝑁})
32	31, 26	eleq2s 2846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ {𝑛, 𝑛} → {𝑛, 𝑛} = {𝑁})
33		preq2 4686	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → {𝑛, 𝑚} = {𝑛, 𝑛})
34	33	eleq2d 2814	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → (𝑁 ∈ {𝑛, 𝑚} ↔ 𝑁 ∈ {𝑛, 𝑛}))
35	33	eqeq1d 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → ({𝑛, 𝑚} = {𝑁} ↔ {𝑛, 𝑛} = {𝑁}))
36	34, 35	imbi12d 344	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑛 → ((𝑁 ∈ {𝑛, 𝑚} → {𝑛, 𝑚} = {𝑁}) ↔ (𝑁 ∈ {𝑛, 𝑛} → {𝑛, 𝑛} = {𝑁})))
37	32, 36	mpbiri 258	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → (𝑁 ∈ {𝑛, 𝑚} → {𝑛, 𝑚} = {𝑁}))
38	37	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 = 𝑛 ∧ 𝑁 ∈ {𝑛, 𝑚}) → {𝑛, 𝑚} = {𝑁})
39	38	olcd 874	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 = 𝑛 ∧ 𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))
40	39	expcom 413	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ {𝑛, 𝑚} → (𝑚 = 𝑛 → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
41	40	3ad2ant3 1135	. . . . . . . . . . . . . 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 32	. . . . . . . . . . . . 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 1197	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑁 ∈ {𝑛, 𝑚})
44		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑚 ≠ 𝑛)
45	44	necomd 2980	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑛 ≠ 𝑚)
46		simpr2 1196	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉))
47		prproe 4856	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ {𝑛, 𝑚} ∧ 𝑛 ≠ 𝑚 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚})
48	43, 45, 46, 47	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚})
49		r19.42v 3161	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ↔ (𝑁 ∈ {𝑛, 𝑚} ∧ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚}))
50	43, 48, 49	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}))
51	50	orcd 873	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 3008	. . . . . . . . . . . 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 1119	. . . . . . . . . . 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 727	. . . . . . . . . 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 406	. . . . . . . . 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 2817	. . . . . . . . . 10 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → (𝑁 ∈ (𝐸‘𝑖) ↔ 𝑁 ∈ {𝑛, 𝑚}))
58		eleq2 2817	. . . . . . . . . . . . 13 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → (𝑣 ∈ (𝐸‘𝑖) ↔ 𝑣 ∈ {𝑛, 𝑚}))
59	57, 58	anbi12d 632	. . . . . . . . . . . 12 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ↔ (𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚})))
60	59	rexbidv 3153	. . . . . . . . . . 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 2733	. . . . . . . . . . 11 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → ((𝐸‘𝑖) = {𝑁} ↔ {𝑛, 𝑚} = {𝑁}))
62	60, 61	orbi12d 918	. . . . . . . . . 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 344	. . . . . . . . 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 247	. . . . . . . 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 3186	. . . . . . 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 47	. . . . . 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 15	. . . . 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 212	. . . 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 3401	. . 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 2776	. 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 2764	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)})

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  {crab 3394   ∖ cdif 3900   ∪ cun 3901  {csn 4577  {cpr 4579  ∪ ciun 4941  dom cdm 5619  Fun wfun 6476  ‘cfv 6482  Vtxcvtx 28941  iEdgciedg 28942  Edgcedg 28992  UHGraphcuhgr 29001  UPGraphcupgr 29025

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-n0 12385  df-xnn0 12458  df-z 12472  df-uz 12736  df-fz 13411  df-hash 14238  df-edg 28993  df-uhgr 29003  df-upgr 29027

This theorem is referenced by:  numedglnl  29089