Theorem numedglnl 29161

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 9215	. . . . . . 7 ⊢ (𝑉 ∈ Fin → (𝑉 ∖ {𝑁}) ∈ Fin)
2	1	adantr 480	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (𝑉 ∖ {𝑁}) ∈ Fin)
3	2	3ad2ant2 1135	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → (𝑉 ∖ {𝑁}) ∈ Fin)
4		dmfi 9375	. . . . . . . . 9 ⊢ (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
5		rabfi 9303	. . . . . . . . 9 ⊢ (dom 𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin)
6	4, 5	syl 17	. . . . . . . 8 ⊢ (𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin)
7	6	adantl 481	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin)
8	7	3ad2ant2 1135	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin)
9	8	adantr 480	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin)
10		notnotb 315	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (𝐸‘𝑖) ↔ ¬ ¬ 𝑁 ∈ (𝐸‘𝑖))
11		notnotb 315	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ (𝐸‘𝑖) ↔ ¬ ¬ 𝑣 ∈ (𝐸‘𝑖))
12		upgruhgr 29119	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
13		edglnl.e	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝐸 = (iEdg‘𝐺)
14	13	uhgrfun 29083	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
15	12, 14	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐺 ∈ UPGraph → Fun 𝐸)
16	13	iedgedg 29067	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Fun 𝐸 ∧ 𝑖 ∈ dom 𝐸) → (𝐸‘𝑖) ∈ (Edg‘𝐺))
17	15, 16	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑖 ∈ dom 𝐸) → (𝐸‘𝑖) ∈ (Edg‘𝐺))
18		edglnl.v	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑉 = (Vtx‘𝐺)
19		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
20	18, 19	upgredg 29154	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺 ∈ UPGraph ∧ (𝐸‘𝑖) ∈ (Edg‘𝐺)) → ∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛})
21	17, 20	syldan 591	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑖 ∈ dom 𝐸) → ∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛})
22	21	ex 412	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐺 ∈ UPGraph → (𝑖 ∈ dom 𝐸 → ∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛}))
23	22	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → (𝑖 ∈ dom 𝐸 → ∃𝑚 ∈ 𝑉 ∃𝑛 ∈ 𝑉 (𝐸‘𝑖) = {𝑚, 𝑛}))
24	23	adantr 480	. . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . 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 4790	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣 ≠ 𝑁)
28		eldifsni 4790	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ (𝑉 ∖ {𝑁}) → 𝑤 ≠ 𝑁)
29		3elpr2eq 4906	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑁 ∈ {𝑚, 𝑛} ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑤 ∈ {𝑚, 𝑛}) ∧ (𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁)) → 𝑣 = 𝑤)
30	29	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) → ((𝑁 ∈ {𝑚, 𝑛} ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑤 ∈ {𝑚, 𝑛}) → 𝑣 = 𝑤))
31	30	3expd 1354	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) → (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → (𝑤 ∈ {𝑚, 𝑛} → 𝑣 = 𝑤))))
32	31	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) → (𝑣 ∈ {𝑚, 𝑛} → (𝑁 ∈ {𝑚, 𝑛} → (𝑤 ∈ {𝑚, 𝑛} → 𝑣 = 𝑤))))
33	32	3imp 1111	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑁 ∈ {𝑚, 𝑛}) → (𝑤 ∈ {𝑚, 𝑛} → 𝑣 = 𝑤))
34	33	con3d 152	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑁 ∈ {𝑚, 𝑛}) → (¬ 𝑣 = 𝑤 → ¬ 𝑤 ∈ {𝑚, 𝑛}))
35	34	3exp 1120	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) → (𝑣 ∈ {𝑚, 𝑛} → (𝑁 ∈ {𝑚, 𝑛} → (¬ 𝑣 = 𝑤 → ¬ 𝑤 ∈ {𝑚, 𝑛}))))
36	35	com24 95	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) → (¬ 𝑣 = 𝑤 → (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛}))))
37	36	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) ∧ ¬ 𝑣 = 𝑤) → (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛})))
38		eleq2 2830	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐸‘𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸‘𝑖) ↔ 𝑁 ∈ {𝑚, 𝑛}))
39		eleq2 2830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐸‘𝑖) = {𝑚, 𝑛} → (𝑣 ∈ (𝐸‘𝑖) ↔ 𝑣 ∈ {𝑚, 𝑛}))
40		eleq2 2830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝐸‘𝑖) = {𝑚, 𝑛} → (𝑤 ∈ (𝐸‘𝑖) ↔ 𝑤 ∈ {𝑚, 𝑛}))
41	40	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐸‘𝑖) = {𝑚, 𝑛} → (¬ 𝑤 ∈ (𝐸‘𝑖) ↔ ¬ 𝑤 ∈ {𝑚, 𝑛}))
42	39, 41	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐸‘𝑖) = {𝑚, 𝑛} → ((𝑣 ∈ (𝐸‘𝑖) → ¬ 𝑤 ∈ (𝐸‘𝑖)) ↔ (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛})))
43	38, 42	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑣 ≠ 𝑁 ∧ 𝑤 ≠ 𝑁) ∧ ¬ 𝑣 = 𝑤) → ((𝐸‘𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸‘𝑖) → (𝑣 ∈ (𝐸‘𝑖) → ¬ 𝑤 ∈ (𝐸‘𝑖)))))
45	44	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 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 3213	. . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 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 15	. . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . . 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 864	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . 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 243	. . . . . . . . . . . . 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 864	. . . . . . . . . . . 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 676	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 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 320	. . . . . . . . . . . . 13 ⊢ (¬ ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ∧ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))) ↔ ¬ (𝑁 ∈ (𝐸‘𝑖) ∧ (𝑣 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))))
62		ianor 984	. . . . . . . . . . . . 13 ⊢ (¬ (𝑁 ∈ (𝐸‘𝑖) ∧ (𝑣 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))) ↔ (¬ 𝑁 ∈ (𝐸‘𝑖) ∨ ¬ (𝑣 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))))
63		ianor 984	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑣 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖)) ↔ (¬ 𝑣 ∈ (𝐸‘𝑖) ∨ ¬ 𝑤 ∈ (𝐸‘𝑖)))
64	63	orbi2i 913	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑁 ∈ (𝐸‘𝑖) ∨ ¬ (𝑣 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))) ↔ (¬ 𝑁 ∈ (𝐸‘𝑖) ∨ (¬ 𝑣 ∈ (𝐸‘𝑖) ∨ ¬ 𝑤 ∈ (𝐸‘𝑖))))
65	61, 62, 64	3bitri 297	. . . . . . . . . . . 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 234	. . . . . . . . . . 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 3146	. . . . . . . . . 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 4316	. . . . . . . . . . . 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 2742	. . . . . . . . . . 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 4388	. . . . . . . . . . 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 275	. . . . . . . . . 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 234	. . . . . . . . 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 412	. . . . . . . 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 864	. . . . . . 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 3202	. . . . . 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 2824	. . . . . . . . 9 ⊢ (𝑣 = 𝑤 → (𝑣 ∈ (𝐸‘𝑖) ↔ 𝑤 ∈ (𝐸‘𝑖)))
77	76	anbi2d 630	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ↔ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))))
78	77	rabbidv 3444	. . . . . . 7 ⊢ (𝑣 = 𝑤 → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} = {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑤 ∈ (𝐸‘𝑖))})
79	78	disjor 5125	. . . . . 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 234	. . . . 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 15858	. . . 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 2743	. . 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 7446	. 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 3146	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin)
85		iunfi 9383	. . . 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 584	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → ∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∈ Fin)
87		rabfi 9303	. . . . . 6 ⊢ (dom 𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} ∈ Fin)
88	4, 87	syl 17	. . . . 5 ⊢ (𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} ∈ Fin)
89	88	adantl 481	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} ∈ Fin)
90	89	3ad2ant2 1135	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} ∈ Fin)
91		fveqeq2 6915	. . . . . . 7 ⊢ (𝑖 = 𝑗 → ((𝐸‘𝑖) = {𝑁} ↔ (𝐸‘𝑗) = {𝑁}))
92	91	elrab 3692	. . . . . 6 ⊢ (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}} ↔ (𝑗 ∈ dom 𝐸 ∧ (𝐸‘𝑗) = {𝑁}))
93		eldifn 4132	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ {𝑁})
94		eleq2 2830	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸‘𝑗) = {𝑁} → (𝑣 ∈ (𝐸‘𝑗) ↔ 𝑣 ∈ {𝑁}))
95	94	notbid 318	. . . . . . . . . . . . . . 15 ⊢ ((𝐸‘𝑗) = {𝑁} → (¬ 𝑣 ∈ (𝐸‘𝑗) ↔ ¬ 𝑣 ∈ {𝑁}))
96	93, 95	imbitrrid 246	. . . . . . . . . . . . . 14 ⊢ ((𝐸‘𝑗) = {𝑁} → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸‘𝑗)))
97	96	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ dom 𝐸 ∧ (𝐸‘𝑗) = {𝑁}) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸‘𝑗)))
98	97	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸‘𝑗) = {𝑁})) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸‘𝑗)))
99	98	imp 406	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸‘𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ 𝑣 ∈ (𝐸‘𝑗))
100	99	intnand 488	. . . . . . . . . 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 488	. . . . . . . . 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 3146	. . . . . . . 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 4995	. . . . . . . . . 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 320	. . . . . . . . 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 3072	. . . . . . . . 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 6906	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝐸‘𝑖) = (𝐸‘𝑗))
107	106	eleq2d 2827	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑁 ∈ (𝐸‘𝑖) ↔ 𝑁 ∈ (𝐸‘𝑗)))
108	106	eleq2d 2827	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑣 ∈ (𝐸‘𝑖) ↔ 𝑣 ∈ (𝐸‘𝑗)))
109	107, 108	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → ((𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖)) ↔ (𝑁 ∈ (𝐸‘𝑗) ∧ 𝑣 ∈ (𝐸‘𝑗))))
110	109	elrab 3692	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ↔ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸‘𝑗) ∧ 𝑣 ∈ (𝐸‘𝑗))))
111	110	notbii 320	. . . . . . . . . 10 ⊢ (¬ 𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ↔ ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸‘𝑗) ∧ 𝑣 ∈ (𝐸‘𝑗))))
112	111	ralbii 3093	. . . . . . . . 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 299	. . . . . . . 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 234	. . . . . . 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 412	. . . . . 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 242	. . . . 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 3145	. . . 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 4451	. . . 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 234	. . 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 14421	. . 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 1373	. 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 29160	. . . 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 1132	. . 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 6910	. 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 2783	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)}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3436   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950  ∅c0 4333  {csn 4626  {cpr 4628  ∪ ciun 4991  Disj wdisj 5110  dom cdm 5685  Fun wfun 6555  ‘cfv 6561  (class class class)co 7431  Fincfn 8985   + caddc 11158  ♯chash 14369  Σcsu 15722  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-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  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  ax-pre-sup 11233

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-rmo 3380  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-disj 5111  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-se 5638  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-isom 6570  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-sup 9482  df-oi 9550  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-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-edg 29065  df-uhgr 29075  df-upgr 29099

This theorem is referenced by:  finsumvtxdg2ssteplem3  29565