Theorem numedglnl 29587

Description: The number of edges incident with a vertex 𝑁 is the number of edges joining 𝑁 with other vertices and the number of loops on 𝑁 in a pseudograph of finite size. (Contributed by AV, 19-Dec-2021.)

Hypotheses

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

Assertion

Ref	Expression
numedglnl	⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → (Σ𝑣 ∈ (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   𝑖,𝐸   𝑣,𝐺

Proof of Theorem numedglnl

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

Step	Hyp	Ref	Expression
1		diffi 9172	. . . . . . 7 ⊢ (𝑉 ∈ Fin → (𝑉 ∖ {𝑁}) ∈ Fin)
2	1	adantr 486	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (𝑉 ∖ {𝑁}) ∈ Fin)
3	2	3ad2ant2 1152	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → (𝑉 ∖ {𝑁}) ∈ Fin)
4		dmfi 9305	. . . . . . . . 9 ⊢ (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
5		rabfi 9244	. . . . . . . . 9 ⊢ (dom 𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin)
6	4, 5	syl 18	. . . . . . . 8 ⊢ (𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin)
7	6	adantl 487	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin)
8	7	3ad2ant2 1152	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin)
9	8	adantr 486	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin)
10		notnotb 318	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (𝐸‘𝑖) ↔ ¬ ¬ 𝑁 ∈ (𝐸‘𝑖))
11		notnotb 318	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ (𝐸‘𝑖) ↔ ¬ ¬ 𝑣 ∈ (𝐸‘𝑖))
12		upgruhgr 29545	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
13		edglnl.e	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝐸 = (iEdg‘𝐺)
14	13	uhgrfun 29509	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
15	12, 14	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐺 ∈ UPGraph → Fun 𝐸)
16	13	iedgedg 29493	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Fun 𝐸 ∧ 𝑖 ∈ dom 𝐸) → (𝐸‘𝑖) ∈ (Edg‘𝐺))
17	15, 16	sylan 592	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑖 ∈ dom 𝐸) → (𝐸‘𝑖) ∈ (Edg‘𝐺))
18		edglnl.v	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑉 = (Vtx‘𝐺)
19		eqid 2762	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
20	18, 19	upgredg 29580	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐸‘𝑖) ∈ (Edg‘𝐺)) → ∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛})
21	17, 20	syldan 603	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑖 ∈ dom 𝐸) → ∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛})
22	21	ex 418	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐺 ∈ UPGraph → (𝑖 ∈ dom 𝐸 → ∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛}))
23	22	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → (𝑖 ∈ dom 𝐸 → ∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛}))
24	23	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) → (𝑖 ∈ dom 𝐸 → ∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛}))
25	24	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) → (𝑖 ∈ dom 𝐸 → ∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛}))
26	25	imp 412	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) → ∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛})
27		eldifsni 4756	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ≠ 𝑁)
28		eldifsni 4756	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ (𝑉 ∖ {𝑁}) → 𝑤 ≠ 𝑁)
29		3elpr2eq 4869	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑁 ∈ {𝑚, 𝑛} ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑤 ∈ {𝑚, 𝑛}) ∧ (𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁)) → 𝑣 = 𝑤)
30	29	expcom 419	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) → ((𝑁 ∈ {𝑚, 𝑛} ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑤 ∈ {𝑚, 𝑛}) → 𝑣 = 𝑤))
31	30	3expd 1372	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) → (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → (𝑤 ∈ {𝑚, 𝑛} → 𝑣 = 𝑤))))
32	31	com23 87	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) → (𝑣 ∈ {𝑚, 𝑛} → (𝑁 ∈ {𝑚, 𝑛} → (𝑤 ∈ {𝑚, 𝑛} → 𝑣 = 𝑤))))
33	32	3imp 1128	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑁 ∈ {𝑚, 𝑛}) → (𝑤 ∈ {𝑚, 𝑛} → 𝑣 = 𝑤))
34	33	con3d 153	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑁 ∈ {𝑚, 𝑛}) → (¬ 𝑣 = 𝑤 → ¬ 𝑤 ∈ {𝑚, 𝑛}))
35	34	3exp 1137	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) → (𝑣 ∈ {𝑚, 𝑛} → (𝑁 ∈ {𝑚, 𝑛} → (¬ 𝑣 = 𝑤 → ¬ 𝑤 ∈ {𝑚, 𝑛}))))
36	35	com24 96	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) → (¬ 𝑣 = 𝑤 → (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛}))))
37	36	imp 412	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) ∧ ¬ 𝑣 = 𝑤) → (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛})))
38		eleq2 2851	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐸‘𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸‘𝑖) ↔ 𝑁 ∈ {𝑚, 𝑛}))
39		eleq2 2851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐸‘𝑖) = {𝑚, 𝑛} → (𝑣 ∈ (𝐸‘𝑖) ↔ 𝑣 ∈ {𝑚, 𝑛}))
40		eleq2 2851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐸‘𝑖) = {𝑚, 𝑛} → (𝑤 ∈ (𝐸‘𝑖) ↔ 𝑤 ∈ {𝑚, 𝑛}))
41	40	notbid 321	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐸‘𝑖) = {𝑚, 𝑛} → (¬ 𝑤 ∈ (𝐸‘𝑖) ↔ ¬ 𝑤 ∈ {𝑚, 𝑛}))
42	39, 41	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐸‘𝑖) = {𝑚, 𝑛} → ((𝑣 ∈ (𝐸‘𝑖) → ¬ 𝑤 ∈ (𝐸‘𝑖)) ↔ (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛})))
43	38, 42	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐸‘𝑖) = {𝑚, 𝑛} → ((𝑁 ∈ (𝐸‘𝑖) → (𝑣 ∈ (𝐸‘𝑖) → ¬ 𝑤 ∈ (𝐸‘𝑖))) ↔ (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛}))))
44	37, 43	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) ∧ ¬ 𝑣 = 𝑤) → ((𝐸‘𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸‘𝑖) → (𝑣 ∈ (𝐸‘𝑖) → ¬ 𝑤 ∈ (𝐸‘𝑖)))))
45	44	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) ∧ ¬ 𝑣 = 𝑤) ∧ (𝑚 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)) → ((𝐸‘𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸‘𝑖) → (𝑣 ∈ (𝐸‘𝑖) → ¬ 𝑤 ∈ (𝐸‘𝑖)))))
46	45	rexlimdvva 3221	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) ∧ ¬ 𝑣 = 𝑤) → (∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸‘𝑖) → (𝑣 ∈ (𝐸‘𝑖) → ¬ 𝑤 ∈ (𝐸‘𝑖)))))
47	46	ex 418	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) → (¬ 𝑣 = 𝑤 → (∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸‘𝑖) → (𝑣 ∈ (𝐸‘𝑖) → ¬ 𝑤 ∈ (𝐸‘𝑖))))))
48	27, 28, 47	syl2an 608	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁})) → (¬ 𝑣 = 𝑤 → (∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸‘𝑖) → (𝑣 ∈ (𝐸‘𝑖) → ¬ 𝑤 ∈ (𝐸‘𝑖))))))
49	48	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) → (¬ 𝑣 = 𝑤 → (∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸‘𝑖) → (𝑣 ∈ (𝐸‘𝑖) → ¬ 𝑤 ∈ (𝐸‘𝑖))))))
50	49	imp 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) → (∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸‘𝑖) → (𝑣 ∈ (𝐸‘𝑖) → ¬ 𝑤 ∈ (𝐸‘𝑖)))))
51	50	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) → (∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸‘𝑖) → (𝑣 ∈ (𝐸‘𝑖) → ¬ 𝑤 ∈ (𝐸‘𝑖)))))
52	26, 51	mpd 16	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) → (𝑁 ∈ (𝐸‘𝑖) → (𝑣 ∈ (𝐸‘𝑖) → ¬ 𝑤 ∈ (𝐸‘𝑖))))
53	52	imp 412	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) ∧ 𝑁 ∈ (𝐸‘𝑖)) → (𝑣 ∈ (𝐸‘𝑖) → ¬ 𝑤 ∈ (𝐸‘𝑖)))
54	11, 53	biimtrrid 246	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) ∧ 𝑁 ∈ (𝐸‘𝑖)) → (¬ ¬ 𝑣 ∈ (𝐸‘𝑖) → ¬ 𝑤 ∈ (𝐸‘𝑖)))
55	54	orrd 877	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) ∧ 𝑁 ∈ (𝐸‘𝑖)) → (¬ 𝑣 ∈ (𝐸‘𝑖) ∨ ¬ 𝑤 ∈ (𝐸‘𝑖)))
56	55	ex 418	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) → (𝑁 ∈ (𝐸‘𝑖) → (¬ 𝑣 ∈ (𝐸‘𝑖) ∨ ¬ 𝑤 ∈ (𝐸‘𝑖))))
57	10, 56	biimtrrid 246	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) → (¬ ¬ 𝑁 ∈ (𝐸‘𝑖) → (¬ 𝑣 ∈ (𝐸‘𝑖) ∨ ¬ 𝑤 ∈ (𝐸‘𝑖))))
58	57	orrd 877	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) → (¬ 𝑁 ∈ (𝐸‘𝑖) ∨ (¬ 𝑣 ∈ (𝐸‘𝑖) ∨ ¬ 𝑤 ∈ (𝐸‘𝑖))))
59		anandi 689	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (𝐸‘𝑖) ∧ (𝑣 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))) ↔ ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∧ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))))
60	59	bicomi 227	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∧ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))) ↔ (𝑁 ∈ (𝐸‘𝑖) ∧ (𝑣 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))))
61	60	notbii 323	. . . . . . . . . . . . 13 ⊢ (¬ ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∧ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))) ↔ ¬ (𝑁 ∈ (𝐸‘𝑖) ∧ (𝑣 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))))
62		ianor 997	. . . . . . . . . . . . 13 ⊢ (¬ (𝑁 ∈ (𝐸‘𝑖) ∧ (𝑣 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))) ↔ (¬ 𝑁 ∈ (𝐸‘𝑖) ∨ ¬ (𝑣 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))))
63		ianor 997	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑣 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖)) ↔ (¬ 𝑣 ∈ (𝐸‘𝑖) ∨ ¬ 𝑤 ∈ (𝐸‘𝑖)))
64	63	orbi2i 926	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑁 ∈ (𝐸‘𝑖) ∨ ¬ (𝑣 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))) ↔ (¬ 𝑁 ∈ (𝐸‘𝑖) ∨ (¬ 𝑣 ∈ (𝐸‘𝑖) ∨ ¬ 𝑤 ∈ (𝐸‘𝑖))))
65	61, 62, 64	3bitri 300	. . . . . . . . . . . 12 ⊢ (¬ ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∧ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))) ↔ (¬ 𝑁 ∈ (𝐸‘𝑖) ∨ (¬ 𝑣 ∈ (𝐸‘𝑖) ∨ ¬ 𝑤 ∈ (𝐸‘𝑖))))
66	58, 65	sylibr 237	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) → ¬ ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∧ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))))
67	66	ralrimiva 3156	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) → ∀𝑖 ∈ dom 𝐸 ¬ ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∧ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))))
68		inrab 4265	. . . . . . . . . . . 12 ⊢ ({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))}) = {𝑖 ∈ dom 𝐸 ∣ ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∧ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖)))}
69	68	eqeq1i 2767	. . . . . . . . . . 11 ⊢ (({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))}) = ∅ ↔ {𝑖 ∈ dom 𝐸 ∣ ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∧ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖)))} = ∅)
70		rabeq0 4341	. . . . . . . . . . 11 ⊢ ({𝑖 ∈ dom 𝐸 ∣ ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∧ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖)))} = ∅ ↔ ∀𝑖 ∈ dom 𝐸 ¬ ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∧ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))))
71	69, 70	bitri 278	. . . . . . . . . 10 ⊢ (({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))}) = ∅ ↔ ∀𝑖 ∈ dom 𝐸 ¬ ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∧ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))))
72	67, 71	sylibr 237	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) → ({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))}) = ∅)
73	72	ex 418	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) → (¬ 𝑣 = 𝑤 → ({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))}) = ∅))
74	73	orrd 877	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) → (𝑣 = 𝑤 ∨ ({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))}) = ∅))
75	74	ralrimivva 3207	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})∀𝑤 ∈ (𝑉 ∖ {𝑁})(𝑣 = 𝑤 ∨ ({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))}) = ∅))
76		eleq1w 2845	. . . . . . . . 9 ⊢ (𝑣 = 𝑤 → (𝑣 ∈ (𝐸‘𝑖) ↔ 𝑤 ∈ (𝐸‘𝑖)))
77	76	anbi2d 642	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ↔ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))))
78	77	rabbidv 3421	. . . . . . 7 ⊢ (𝑣 = 𝑤 → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} = {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))})
79	78	disjor 5089	. . . . . 6 ⊢ (Disj 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁})∀𝑤 ∈ (𝑉 ∖ {𝑁})(𝑣 = 𝑤 ∨ ({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))}) = ∅))
80	75, 79	sylibr 237	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → Disj 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))})
81	3, 9, 80	hashiun 15911	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → (♯‘∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))}) = Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))}))
82	81	eqcomd 2768	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))}) = (♯‘∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))}))
83	82	oveq1d 7431	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})) = ((♯‘∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})))
84	9	ralrimiva 3156	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin)
85		iunfi 9313	. . . 4 ⊢ (((𝑉 ∖ {𝑁}) ∈ Fin ∧ ∀𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin) → ∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin)
86	3, 84, 85	syl2anc 596	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → ∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin)
87		rabfi 9244	. . . . . 6 ⊢ (dom 𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} ∈ Fin)
88	4, 87	syl 18	. . . . 5 ⊢ (𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} ∈ Fin)
89	88	adantl 487	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} ∈ Fin)
90	89	3ad2ant2 1152	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} ∈ Fin)
91		fveqeq2 6891	. . . . . . 7 ⊢ (𝑖 = 𝑗 → ((𝐸‘𝑖) = {𝑁} ↔ (𝐸‘𝑗) = {𝑁}))
92	91	elrab 3648	. . . . . 6 ⊢ (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} ↔ (𝑗 ∈ dom 𝐸 ∧ (𝐸‘𝑗) = {𝑁}))
93		eldifn 4082	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ {𝑁})
94		eleq2 2851	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸‘𝑗) = {𝑁} → (𝑣 ∈ (𝐸‘𝑗) ↔ 𝑣 ∈ {𝑁}))
95	94	notbid 321	. . . . . . . . . . . . . . 15 ⊢ ((𝐸‘𝑗) = {𝑁} → (¬ 𝑣 ∈ (𝐸‘𝑗) ↔ ¬ 𝑣 ∈ {𝑁}))
96	93, 95	imbitrrid 249	. . . . . . . . . . . . . 14 ⊢ ((𝐸‘𝑗) = {𝑁} → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸‘𝑗)))
97	96	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ dom 𝐸 ∧ (𝐸‘𝑗) = {𝑁}) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸‘𝑗)))
98	97	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸‘𝑗) = {𝑁})) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸‘𝑗)))
99	98	imp 412	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸‘𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ 𝑣 ∈ (𝐸‘𝑗))
100	99	intnand 494	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸‘𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ (𝑁 ∈ (𝐸‘𝑗) ∧ 𝑣 ∈ (𝐸‘𝑗)))
101	100	intnand 494	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸‘𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸‘𝑗) ∧ 𝑣 ∈ (𝐸‘𝑗))))
102	101	ralrimiva 3156	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸‘𝑗) = {𝑁})) → ∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸‘𝑗) ∧ 𝑣 ∈ (𝐸‘𝑗))))
103		eliun 4958	. . . . . . . . . 10 ⊢ (𝑗 ∈ ∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ↔ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))})
104	103	notbii 323	. . . . . . . . 9 ⊢ (¬ 𝑗 ∈ ∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ↔ ¬ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))})
105		ralnex 3090	. . . . . . . . 9 ⊢ (∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ 𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ↔ ¬ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))})
106		fveq2 6882	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝐸‘𝑖) = (𝐸‘𝑗))
107	106	eleq2d 2848	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑁 ∈ (𝐸‘𝑖) ↔ 𝑁 ∈ (𝐸‘𝑗)))
108	106	eleq2d 2848	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑣 ∈ (𝐸‘𝑖) ↔ 𝑣 ∈ (𝐸‘𝑗)))
109	107, 108	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ↔ (𝑁 ∈ (𝐸‘𝑗) ∧ 𝑣 ∈ (𝐸‘𝑗))))
110	109	elrab 3648	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ↔ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸‘𝑗) ∧ 𝑣 ∈ (𝐸‘𝑗))))
111	110	notbii 323	. . . . . . . . . 10 ⊢ (¬ 𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ↔ ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸‘𝑗) ∧ 𝑣 ∈ (𝐸‘𝑗))))
112	111	ralbii 3110	. . . . . . . . 9 ⊢ (∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ 𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸‘𝑗) ∧ 𝑣 ∈ (𝐸‘𝑗))))
113	104, 105, 112	3bitr2i 302	. . . . . . . 8 ⊢ (¬ 𝑗 ∈ ∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸‘𝑗) ∧ 𝑣 ∈ (𝐸‘𝑗))))
114	102, 113	sylibr 237	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸‘𝑗) = {𝑁})) → ¬ 𝑗 ∈ ∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))})
115	114	ex 418	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → ((𝑗 ∈ dom 𝐸 ∧ (𝐸‘𝑗) = {𝑁}) → ¬ 𝑗 ∈ ∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))}))
116	92, 115	biimtrid 245	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} → ¬ 𝑗 ∈ ∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))}))
117	116	ralrimiv 3155	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → ∀𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} ¬ 𝑗 ∈ ∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))})
118		disjr 4407	. . . 4 ⊢ ((∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}) = ∅ ↔ ∀𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} ¬ 𝑗 ∈ ∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))})
119	117, 118	sylibr 237	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → (∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}) = ∅)
120		hashun 14448	. . 3 ⊢ ((∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin ∧ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} ∈ Fin ∧ (∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}) = ∅) → (♯‘(∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})) = ((♯‘∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})))
121	86, 90, 119, 120	syl3anc 1398	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → (♯‘(∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})) = ((♯‘∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})))
122	18, 13	edglnl 29586	. . . 4 ⊢ ((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)})
123	122	3adant2 1149	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 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)})
124	123	fveq2d 6886	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 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)}))
125	83, 121, 124	3eqtr2d 2803	1 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → (Σ𝑣 ∈ (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:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3414   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901  ∅c0 4282  {csn 4587  {cpr 4589  ∪ ciun 4954  Disj wdisj 5074  dom cdm 5659  Fun wfun 6531  ‘cfv 6537  (class class class)co 7416  Fincfn 8955   + caddc 11130  ♯chash 14396  Σcsu 15775  Vtxcvtx 29439  iEdgciedg 29440  Edgcedg 29490  UHGraphcuhgr 29499  UPGraphcupgr 29523

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-disj 5075  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-oadd 8462  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9415  df-oi 9485  df-dju 9909  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-n0 12532  df-xnn0 12605  df-z 12619  df-uz 12891  df-rp 13045  df-fz 13564  df-fzo 13712  df-seq 14068  df-exp 14128  df-hash 14397  df-cj 15188  df-re 15189  df-im 15190  df-sqrt 15324  df-abs 15325  df-clim 15577  df-sum 15776  df-edg 29491  df-uhgr 29501  df-upgr 29525

This theorem is used by:  finsumvtxdg2ssteplem3  29993