Theorem edglnl 29160

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 5052	. . . 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 4167	. 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 4315	. . 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 3151	. . . . . . 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 4660	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ {𝑁})
9	8	ad2antlr 727	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → 𝑁 ∈ {𝑁})
10		eleq2 2830	. . . . . . 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 860	. . . . 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 29119	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
14		edglnl.e	. . . . . . . . . 10 ⊢ 𝐸 = (iEdg‘𝐺)
15	14	uhgrfun 29083	. . . . . . . . 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 29067	. . . . . . 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 2737	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
22	20, 21	upgredg 29154	. . . . . . . . 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 4639	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {𝑛} = {𝑛, 𝑛}
26	25	eqcomi 2746	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑛, 𝑛} = {𝑛}
27		elsni 4643	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ {𝑛} → 𝑁 = 𝑛)
28		sneq 4636	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 = 𝑛 → {𝑁} = {𝑛})
29	28	eqcomd 2743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 = 𝑛 → {𝑛} = {𝑁})
30	27, 29	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ {𝑛} → {𝑛} = {𝑁})
31	26, 30	eqtrid 2789	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ {𝑛} → {𝑛, 𝑛} = {𝑁})
32	31, 26	eleq2s 2859	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ {𝑛, 𝑛} → {𝑛, 𝑛} = {𝑁})
33		preq2 4734	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → {𝑛, 𝑚} = {𝑛, 𝑛})
34	33	eleq2d 2827	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → (𝑁 ∈ {𝑛, 𝑚} ↔ 𝑁 ∈ {𝑛, 𝑛}))
35	33	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . 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 875	. . . . . . . . . . . . . . . 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 1136	. . . . . . . . . . . . . 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 2996	. . . . . . . . . . . . . . . . 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 4905	. . . . . . . . . . . . . . . . 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 3191	. . . . . . . . . . . . . . . 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 874	. . . . . . . . . . . . . 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 3025	. . . . . . . . . . . 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 1120	. . . . . . . . . . 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 2830	. . . . . . . . . 10 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → (𝑁 ∈ (𝐸‘𝑖) ↔ 𝑁 ∈ {𝑛, 𝑚}))
58		eleq2 2830	. . . . . . . . . . . . 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 3179	. . . . . . . . . . 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 2741	. . . . . . . . . . 11 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → ((𝐸‘𝑖) = {𝑁} ↔ {𝑛, 𝑚} = {𝑁}))
62	60, 61	orbi12d 919	. . . . . . . . . 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 3213	. . . . . . 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 3443	. . 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 2789	. 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 2777	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 848   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070  {crab 3436   ∖ cdif 3948   ∪ cun 3949  {csn 4626  {cpr 4628  ∪ ciun 4991  dom cdm 5685  Fun wfun 6555  ‘cfv 6561  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064  UHGraphcuhgr 29073  UPGraphcupgr 29097

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370  df-edg 29065  df-uhgr 29075  df-upgr 29099

This theorem is referenced by:  numedglnl  29161