Theorem edglnl 27035

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 4941	. . . 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 4067	. 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 4208	. . 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 486	. . . . . . . 8 ⊢ ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) → 𝑁 ∈ (𝐸‘𝑖))
6	5	rexlimivw 3206	. . . . . . 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 4556	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ {𝑁})
9	8	ad2antlr 726	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → 𝑁 ∈ {𝑁})
10		eleq2 2840	. . . . . . 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 856	. . . . 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 26994	. . . . . . . . 9 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
14		edglnl.e	. . . . . . . . . 10 ⊢ 𝐸 = (iEdg‘𝐺)
15	14	uhgrfun 26958	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
16	13, 15	syl 17	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → Fun 𝐸)
17	16	adantr 484	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → Fun 𝐸)
18	14	iedgedg 26942	. . . . . . 7 ⊢ ((Fun 𝐸 ∧ 𝑖 ∈ dom 𝐸) → (𝐸‘𝑖) ∈ (Edg‘𝐺))
19	17, 18	sylan 583	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → (𝐸‘𝑖) ∈ (Edg‘𝐺))
20		edglnl.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
21		eqid 2758	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
22	20, 21	upgredg 27029	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐸‘𝑖) ∈ (Edg‘𝐺)) → ∃𝑛 ∈ 𝑉 ∃𝑚 ∈ 𝑉 (𝐸‘𝑖) = {𝑛, 𝑚})
23	22	ex 416	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → ((𝐸‘𝑖) ∈ (Edg‘𝐺) → ∃𝑛 ∈ 𝑉 ∃𝑚 ∈ 𝑉 (𝐸‘𝑖) = {𝑛, 𝑚}))
24	23	ad2antrr 725	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝐸‘𝑖) ∈ (Edg‘𝐺) → ∃𝑛 ∈ 𝑉 ∃𝑚 ∈ 𝑉 (𝐸‘𝑖) = {𝑛, 𝑚}))
25		dfsn2 4535	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {𝑛} = {𝑛, 𝑛}
26	25	eqcomi 2767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑛, 𝑛} = {𝑛}
27		elsni 4539	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ {𝑛} → 𝑁 = 𝑛)
28		sneq 4532	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 = 𝑛 → {𝑁} = {𝑛})
29	28	eqcomd 2764	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 = 𝑛 → {𝑛} = {𝑁})
30	27, 29	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ {𝑛} → {𝑛} = {𝑁})
31	26, 30	syl5eq 2805	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ {𝑛} → {𝑛, 𝑛} = {𝑁})
32	31, 26	eleq2s 2870	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ {𝑛, 𝑛} → {𝑛, 𝑛} = {𝑁})
33		preq2 4627	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑛 → {𝑛, 𝑚} = {𝑛, 𝑛})
34	33	eleq2d 2837	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → (𝑁 ∈ {𝑛, 𝑚} ↔ 𝑁 ∈ {𝑛, 𝑛}))
35	33	eqeq1d 2760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → ({𝑛, 𝑚} = {𝑁} ↔ {𝑛, 𝑛} = {𝑁}))
36	34, 35	imbi12d 348	. . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 = 𝑛 ∧ 𝑁 ∈ {𝑛, 𝑚}) → {𝑛, 𝑚} = {𝑁})
39	38	olcd 871	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 = 𝑛 ∧ 𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))
40	39	expcom 417	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ {𝑛, 𝑚} → (𝑚 = 𝑛 → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
41	40	3ad2ant3 1132	. . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑁 ∈ {𝑛, 𝑚})
44		simpl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑚 ≠ 𝑛)
45	44	necomd 3006	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑛 ≠ 𝑚)
46		simpr2 1192	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉))
47		prproe 4796	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ {𝑛, 𝑚} ∧ 𝑛 ≠ 𝑚 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚})
48	43, 45, 46, 47	syl3anc 1368	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚})
49		r19.42v 3268	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ↔ (𝑁 ∈ {𝑛, 𝑚} ∧ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚}))
50	43, 48, 49	sylanbrc 586	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ≠ 𝑛 ∧ (𝑁 ∈ 𝑉 ∧ (𝑛 ∈ 𝑉 ∧ 𝑚 ∈ 𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}))
51	50	orcd 870	. . . . . . . . . . . . . 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 416	. . . . . . . . . . . . 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 3034	. . . . . . . . . . . 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 1116	. . . . . . . . . . 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 726	. . . . . . . . . 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 410	. . . . . . . . 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 2840	. . . . . . . . . 10 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → (𝑁 ∈ (𝐸‘𝑖) ↔ 𝑁 ∈ {𝑛, 𝑚}))
58		eleq2 2840	. . . . . . . . . . . . 13 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → (𝑣 ∈ (𝐸‘𝑖) ↔ 𝑣 ∈ {𝑛, 𝑚}))
59	57, 58	anbi12d 633	. . . . . . . . . . . 12 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ↔ (𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚})))
60	59	rexbidv 3221	. . . . . . . . . . 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 2762	. . . . . . . . . . 11 ⊢ ((𝐸‘𝑖) = {𝑛, 𝑚} → ((𝐸‘𝑖) = {𝑁} ↔ {𝑛, 𝑚} = {𝑁}))
62	60, 61	orbi12d 916	. . . . . . . . . 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 348	. . . . . . . . 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 3218	. . . . . . 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 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 3390	. . 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	syl5eq 2805	. 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 2793	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 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∃wrex 3071  {crab 3074   ∖ cdif 3855   ∪ cun 3856  {csn 4522  {cpr 4524  ∪ ciun 4883  dom cdm 5524  Fun wfun 6329  ‘cfv 6335  Vtxcvtx 26888  iEdgciedg 26889  Edgcedg 26939  UHGraphcuhgr 26948  UPGraphcupgr 26972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-2o 8113  df-oadd 8116  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-dju 9363  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-n0 11935  df-xnn0 12007  df-z 12021  df-uz 12283  df-fz 12940  df-hash 13741  df-edg 26940  df-uhgr 26950  df-upgr 26974

This theorem is referenced by:  numedglnl  27036