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Theorem numedglnl 27235
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) ∧ 𝑁𝑉) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})) = (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))
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.
StepHypRef Expression
1 diffi 8906 . . . . . . 7 (𝑉 ∈ Fin → (𝑉 ∖ {𝑁}) ∈ Fin)
21adantr 484 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (𝑉 ∖ {𝑁}) ∈ Fin)
323ad2ant2 1136 . . . . 5 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (𝑉 ∖ {𝑁}) ∈ Fin)
4 dmfi 8954 . . . . . . . . 9 (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
5 rabfi 8900 . . . . . . . . 9 (dom 𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
64, 5syl 17 . . . . . . . 8 (𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
76adantl 485 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
873ad2ant2 1136 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
98adantr 484 . . . . 5 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
10 notnotb 318 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝐸𝑖) ↔ ¬ ¬ 𝑁 ∈ (𝐸𝑖))
11 notnotb 318 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ (𝐸𝑖) ↔ ¬ ¬ 𝑣 ∈ (𝐸𝑖))
12 upgruhgr 27193 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
13 edglnl.e . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐸 = (iEdg‘𝐺)
1413uhgrfun 27157 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺 ∈ UHGraph → Fun 𝐸)
1512, 14syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐺 ∈ UPGraph → Fun 𝐸)
1613iedgedg 27141 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Fun 𝐸𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
1715, 16sylan 583 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ UPGraph ∧ 𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
18 edglnl.v . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑉 = (Vtx‘𝐺)
19 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Edg‘𝐺) = (Edg‘𝐺)
2018, 19upgredg 27228 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ UPGraph ∧ (𝐸𝑖) ∈ (Edg‘𝐺)) → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛})
2117, 20syldan 594 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ UPGraph ∧ 𝑖 ∈ dom 𝐸) → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛})
2221ex 416 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 ∈ UPGraph → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
23223ad2ant1 1135 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
2423adantr 484 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
2524adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
2625imp 410 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛})
27 eldifsni 4703 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣𝑁)
28 eldifsni 4703 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ (𝑉 ∖ {𝑁}) → 𝑤𝑁)
29 3elpr2eq 4818 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑁 ∈ {𝑚, 𝑛} ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑤 ∈ {𝑚, 𝑛}) ∧ (𝑣𝑁𝑤𝑁)) → 𝑣 = 𝑤)
3029expcom 417 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑣𝑁𝑤𝑁) → ((𝑁 ∈ {𝑚, 𝑛} ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑤 ∈ {𝑚, 𝑛}) → 𝑣 = 𝑤))
31303expd 1355 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑣𝑁𝑤𝑁) → (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → (𝑤 ∈ {𝑚, 𝑛} → 𝑣 = 𝑤))))
3231com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑣𝑁𝑤𝑁) → (𝑣 ∈ {𝑚, 𝑛} → (𝑁 ∈ {𝑚, 𝑛} → (𝑤 ∈ {𝑚, 𝑛} → 𝑣 = 𝑤))))
33323imp 1113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑣𝑁𝑤𝑁) ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑁 ∈ {𝑚, 𝑛}) → (𝑤 ∈ {𝑚, 𝑛} → 𝑣 = 𝑤))
3433con3d 155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑣𝑁𝑤𝑁) ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑁 ∈ {𝑚, 𝑛}) → (¬ 𝑣 = 𝑤 → ¬ 𝑤 ∈ {𝑚, 𝑛}))
35343exp 1121 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑣𝑁𝑤𝑁) → (𝑣 ∈ {𝑚, 𝑛} → (𝑁 ∈ {𝑚, 𝑛} → (¬ 𝑣 = 𝑤 → ¬ 𝑤 ∈ {𝑚, 𝑛}))))
3635com24 95 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑣𝑁𝑤𝑁) → (¬ 𝑣 = 𝑤 → (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛}))))
3736imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) → (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛})))
38 eleq2 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ {𝑚, 𝑛}))
39 eleq2 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐸𝑖) = {𝑚, 𝑛} → (𝑣 ∈ (𝐸𝑖) ↔ 𝑣 ∈ {𝑚, 𝑛}))
40 eleq2 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐸𝑖) = {𝑚, 𝑛} → (𝑤 ∈ (𝐸𝑖) ↔ 𝑤 ∈ {𝑚, 𝑛}))
4140notbid 321 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐸𝑖) = {𝑚, 𝑛} → (¬ 𝑤 ∈ (𝐸𝑖) ↔ ¬ 𝑤 ∈ {𝑚, 𝑛}))
4239, 41imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐸𝑖) = {𝑚, 𝑛} → ((𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)) ↔ (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛})))
4338, 42imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐸𝑖) = {𝑚, 𝑛} → ((𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))) ↔ (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛}))))
4437, 43syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) → ((𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))))
4544adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) ∧ (𝑚𝑉𝑛𝑉)) → ((𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))))
4645rexlimdvva 3213 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))))
4746ex 416 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑣𝑁𝑤𝑁) → (¬ 𝑣 = 𝑤 → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))))))
4827, 28, 47syl2an 599 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁})) → (¬ 𝑣 = 𝑤 → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))))))
4948adantl 485 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) → (¬ 𝑣 = 𝑤 → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))))))
5049imp 410 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))))
5150adantr 484 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))))
5226, 51mpd 15 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))))
5352imp 410 . . . . . . . . . . . . . . . . 17 ((((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) ∧ 𝑁 ∈ (𝐸𝑖)) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))
5411, 53syl5bir 246 . . . . . . . . . . . . . . . 16 ((((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) ∧ 𝑁 ∈ (𝐸𝑖)) → (¬ ¬ 𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))
5554orrd 863 . . . . . . . . . . . . . . 15 ((((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) ∧ 𝑁 ∈ (𝐸𝑖)) → (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖)))
5655ex 416 . . . . . . . . . . . . . 14 (((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) → (𝑁 ∈ (𝐸𝑖) → (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖))))
5710, 56syl5bir 246 . . . . . . . . . . . . 13 (((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) → (¬ ¬ 𝑁 ∈ (𝐸𝑖) → (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖))))
5857orrd 863 . . . . . . . . . . . 12 (((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) → (¬ 𝑁 ∈ (𝐸𝑖) ∨ (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖))))
59 anandi 676 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
6059bicomi 227 . . . . . . . . . . . . . 14 (((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
6160notbii 323 . . . . . . . . . . . . 13 (¬ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ ¬ (𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
62 ianor 982 . . . . . . . . . . . . 13 (¬ (𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (¬ 𝑁 ∈ (𝐸𝑖) ∨ ¬ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
63 ianor 982 . . . . . . . . . . . . . 14 (¬ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖)) ↔ (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖)))
6463orbi2i 913 . . . . . . . . . . . . 13 ((¬ 𝑁 ∈ (𝐸𝑖) ∨ ¬ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (¬ 𝑁 ∈ (𝐸𝑖) ∨ (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖))))
6561, 62, 643bitri 300 . . . . . . . . . . . 12 (¬ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (¬ 𝑁 ∈ (𝐸𝑖) ∨ (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖))))
6658, 65sylibr 237 . . . . . . . . . . 11 (((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) ∧ 𝑖 ∈ dom 𝐸) → ¬ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
6766ralrimiva 3105 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) → ∀𝑖 ∈ dom 𝐸 ¬ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
68 inrab 4221 . . . . . . . . . . . 12 ({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))}) = {𝑖 ∈ dom 𝐸 ∣ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖)))}
6968eqeq1i 2742 . . . . . . . . . . 11 (({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))}) = ∅ ↔ {𝑖 ∈ dom 𝐸 ∣ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖)))} = ∅)
70 rabeq0 4299 . . . . . . . . . . 11 ({𝑖 ∈ dom 𝐸 ∣ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖)))} = ∅ ↔ ∀𝑖 ∈ dom 𝐸 ¬ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
7169, 70bitri 278 . . . . . . . . . 10 (({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))}) = ∅ ↔ ∀𝑖 ∈ dom 𝐸 ¬ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
7267, 71sylibr 237 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) → ({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))}) = ∅)
7372ex 416 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) → (¬ 𝑣 = 𝑤 → ({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))}) = ∅))
7473orrd 863 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) → (𝑣 = 𝑤 ∨ ({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))}) = ∅))
7574ralrimivva 3112 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})∀𝑤 ∈ (𝑉 ∖ {𝑁})(𝑣 = 𝑤 ∨ ({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))}) = ∅))
76 eleq1w 2820 . . . . . . . . 9 (𝑣 = 𝑤 → (𝑣 ∈ (𝐸𝑖) ↔ 𝑤 ∈ (𝐸𝑖)))
7776anbi2d 632 . . . . . . . 8 (𝑣 = 𝑤 → ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
7877rabbidv 3390 . . . . . . 7 (𝑣 = 𝑤 → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} = {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))})
7978disjor 5033 . . . . . 6 (Disj 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁})∀𝑤 ∈ (𝑉 ∖ {𝑁})(𝑣 = 𝑤 ∨ ({𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))}) = ∅))
8075, 79sylibr 237 . . . . 5 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → Disj 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
813, 9, 80hashiun 15386 . . . 4 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (♯‘ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))}) = Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))}))
8281eqcomd 2743 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))}) = (♯‘ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))}))
8382oveq1d 7228 . 2 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})) = ((♯‘ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})))
849ralrimiva 3105 . . . 4 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
85 iunfi 8964 . . . 4 (((𝑉 ∖ {𝑁}) ∈ Fin ∧ ∀𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin) → 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
863, 84, 85syl2anc 587 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
87 rabfi 8900 . . . . . 6 (dom 𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
884, 87syl 17 . . . . 5 (𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
8988adantl 485 . . . 4 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
90893ad2ant2 1136 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
91 fveqeq2 6726 . . . . . . 7 (𝑖 = 𝑗 → ((𝐸𝑖) = {𝑁} ↔ (𝐸𝑗) = {𝑁}))
9291elrab 3602 . . . . . 6 (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ↔ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁}))
93 eldifn 4042 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ {𝑁})
94 eleq2 2826 . . . . . . . . . . . . . . . 16 ((𝐸𝑗) = {𝑁} → (𝑣 ∈ (𝐸𝑗) ↔ 𝑣 ∈ {𝑁}))
9594notbid 321 . . . . . . . . . . . . . . 15 ((𝐸𝑗) = {𝑁} → (¬ 𝑣 ∈ (𝐸𝑗) ↔ ¬ 𝑣 ∈ {𝑁}))
9693, 95syl5ibr 249 . . . . . . . . . . . . . 14 ((𝐸𝑗) = {𝑁} → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸𝑗)))
9796adantl 485 . . . . . . . . . . . . 13 ((𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁}) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸𝑗)))
9897adantl 485 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸𝑗)))
9998imp 410 . . . . . . . . . . 11 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ 𝑣 ∈ (𝐸𝑗))
10099intnand 492 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗)))
101100intnand 492 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
102101ralrimiva 3105 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) → ∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
103 eliun 4908 . . . . . . . . . 10 (𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
104103notbii 323 . . . . . . . . 9 𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ¬ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
105 ralnex 3158 . . . . . . . . 9 (∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ 𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ¬ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
106 fveq2 6717 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝐸𝑖) = (𝐸𝑗))
107106eleq2d 2823 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ (𝐸𝑗)))
108106eleq2d 2823 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑣 ∈ (𝐸𝑖) ↔ 𝑣 ∈ (𝐸𝑗)))
109107, 108anbi12d 634 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
110109elrab 3602 . . . . . . . . . . 11 (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
111110notbii 323 . . . . . . . . . 10 𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
112111ralbii 3088 . . . . . . . . 9 (∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ 𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
113104, 105, 1123bitr2i 302 . . . . . . . 8 𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
114102, 113sylibr 237 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) → ¬ 𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
115114ex 416 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁}) → ¬ 𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))}))
11692, 115syl5bi 245 . . . . 5 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} → ¬ 𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))}))
117116ralrimiv 3104 . . . 4 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → ∀𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ¬ 𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
118 disjr 4364 . . . 4 (( 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}) = ∅ ↔ ∀𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ¬ 𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
119117, 118sylibr 237 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → ( 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}) = ∅)
120 hashun 13949 . . 3 (( 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin ∧ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin ∧ ( 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∩ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}) = ∅) → (♯‘( 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})) = ((♯‘ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})))
12186, 90, 119, 120syl3anc 1373 . 2 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (♯‘( 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})) = ((♯‘ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})))
12218, 13edglnl 27234 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}) = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
1231223adant2 1133 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → ( 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}) = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
124123fveq2d 6721 . 2 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (♯‘( 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})) = (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))
12583, 121, 1243eqtr2d 2783 1 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})) = (♯‘{𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 847  w3a 1089   = wceq 1543  wcel 2110  wne 2940  wral 3061  wrex 3062  {crab 3065  cdif 3863  cun 3864  cin 3865  c0 4237  {csn 4541  {cpr 4543   ciun 4904  Disj wdisj 5018  dom cdm 5551  Fun wfun 6374  cfv 6380  (class class class)co 7213  Fincfn 8626   + caddc 10732  chash 13896  Σcsu 15249  Vtxcvtx 27087  iEdgciedg 27088  Edgcedg 27138  UHGraphcuhgr 27147  UPGraphcupgr 27171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-disj 5019  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-oadd 8206  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-sup 9058  df-oi 9126  df-dju 9517  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-xnn0 12163  df-z 12177  df-uz 12439  df-rp 12587  df-fz 13096  df-fzo 13239  df-seq 13575  df-exp 13636  df-hash 13897  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-clim 15049  df-sum 15250  df-edg 27139  df-uhgr 27149  df-upgr 27173
This theorem is referenced by:  finsumvtxdg2ssteplem3  27635
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