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Theorem numedglnl 28999
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 9200 . . . . . . 7 (𝑉 ∈ Fin → (𝑉 ∖ {𝑁}) ∈ Fin)
21adantr 479 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (𝑉 ∖ {𝑁}) ∈ Fin)
323ad2ant2 1131 . . . . 5 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (𝑉 ∖ {𝑁}) ∈ Fin)
4 dmfi 9352 . . . . . . . . 9 (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
5 rabfi 9290 . . . . . . . . 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 480 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
873ad2ant2 1131 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
98adantr 479 . . . . 5 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
10 notnotb 314 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝐸𝑖) ↔ ¬ ¬ 𝑁 ∈ (𝐸𝑖))
11 notnotb 314 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ (𝐸𝑖) ↔ ¬ ¬ 𝑣 ∈ (𝐸𝑖))
12 upgruhgr 28957 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
13 edglnl.e . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐸 = (iEdg‘𝐺)
1413uhgrfun 28921 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺 ∈ UHGraph → Fun 𝐸)
1512, 14syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐺 ∈ UPGraph → Fun 𝐸)
1613iedgedg 28905 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Fun 𝐸𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
1715, 16sylan 578 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ UPGraph ∧ 𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
18 edglnl.v . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑉 = (Vtx‘𝐺)
19 eqid 2725 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Edg‘𝐺) = (Edg‘𝐺)
2018, 19upgredg 28992 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ UPGraph ∧ (𝐸𝑖) ∈ (Edg‘𝐺)) → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛})
2117, 20syldan 589 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ UPGraph ∧ 𝑖 ∈ dom 𝐸) → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛})
2221ex 411 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 ∈ UPGraph → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
23223ad2ant1 1130 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
2423adantr 479 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
2524adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
2625imp 405 . . . . . . . . . . . . . . . . . . 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 4789 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣𝑁)
28 eldifsni 4789 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ (𝑉 ∖ {𝑁}) → 𝑤𝑁)
29 3elpr2eq 4902 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑁 ∈ {𝑚, 𝑛} ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑤 ∈ {𝑚, 𝑛}) ∧ (𝑣𝑁𝑤𝑁)) → 𝑣 = 𝑤)
3029expcom 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑣𝑁𝑤𝑁) → ((𝑁 ∈ {𝑚, 𝑛} ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑤 ∈ {𝑚, 𝑛}) → 𝑣 = 𝑤))
31303expd 1350 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑣𝑁𝑤𝑁) ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑁 ∈ {𝑚, 𝑛}) → (𝑤 ∈ {𝑚, 𝑛} → 𝑣 = 𝑤))
3433con3d 152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑣𝑁𝑤𝑁) ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑁 ∈ {𝑚, 𝑛}) → (¬ 𝑣 = 𝑤 → ¬ 𝑤 ∈ {𝑚, 𝑛}))
35343exp 1116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) → (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛})))
38 eleq2 2814 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ {𝑚, 𝑛}))
39 eleq2 2814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐸𝑖) = {𝑚, 𝑛} → (𝑣 ∈ (𝐸𝑖) ↔ 𝑣 ∈ {𝑚, 𝑛}))
40 eleq2 2814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐸𝑖) = {𝑚, 𝑛} → (𝑤 ∈ (𝐸𝑖) ↔ 𝑤 ∈ {𝑚, 𝑛}))
4140notbid 317 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐸𝑖) = {𝑚, 𝑛} → (¬ 𝑤 ∈ (𝐸𝑖) ↔ ¬ 𝑤 ∈ {𝑚, 𝑛}))
4239, 41imbi12d 343 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐸𝑖) = {𝑚, 𝑛} → ((𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)) ↔ (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛})))
4338, 42imbi12d 343 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐸𝑖) = {𝑚, 𝑛} → ((𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))) ↔ (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛}))))
4437, 43syl5ibrcom 246 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) → ((𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))))
4544adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) ∧ (𝑚𝑉𝑛𝑉)) → ((𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))))
4645rexlimdvva 3202 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))))
4746ex 411 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑣𝑁𝑤𝑁) → (¬ 𝑣 = 𝑤 → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))))))
4827, 28, 47syl2an 594 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁})) → (¬ 𝑣 = 𝑤 → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))))))
4948adantl 480 . . . . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . . . 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, 53biimtrrid 242 . . . . . . . . . . . . . . . 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 861 . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . 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, 56biimtrrid 242 . . . . . . . . . . . . 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 861 . . . . . . . . . . . 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 674 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
6059bicomi 223 . . . . . . . . . . . . . 14 (((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
6160notbii 319 . . . . . . . . . . . . 13 (¬ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ ¬ (𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
62 ianor 979 . . . . . . . . . . . . 13 (¬ (𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (¬ 𝑁 ∈ (𝐸𝑖) ∨ ¬ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
63 ianor 979 . . . . . . . . . . . . . 14 (¬ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖)) ↔ (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖)))
6463orbi2i 910 . . . . . . . . . . . . 13 ((¬ 𝑁 ∈ (𝐸𝑖) ∨ ¬ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (¬ 𝑁 ∈ (𝐸𝑖) ∨ (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖))))
6561, 62, 643bitri 296 . . . . . . . . . . . 12 (¬ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (¬ 𝑁 ∈ (𝐸𝑖) ∨ (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖))))
6658, 65sylibr 233 . . . . . . . . . . 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 3136 . . . . . . . . . 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 4301 . . . . . . . . . . . 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 2730 . . . . . . . . . . 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 4380 . . . . . . . . . . 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 274 . . . . . . . . . 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 233 . . . . . . . . 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 411 . . . . . . . 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 861 . . . . . . 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 3191 . . . . . 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 2808 . . . . . . . . 9 (𝑣 = 𝑤 → (𝑣 ∈ (𝐸𝑖) ↔ 𝑤 ∈ (𝐸𝑖)))
7776anbi2d 628 . . . . . . . 8 (𝑣 = 𝑤 → ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
7877rabbidv 3427 . . . . . . 7 (𝑣 = 𝑤 → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} = {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))})
7978disjor 5123 . . . . . 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 233 . . . . 5 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → Disj 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
813, 9, 80hashiun 15798 . . . 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 2731 . . 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 7430 . 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 3136 . . . 4 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
85 iunfi 9362 . . . 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 582 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
87 rabfi 9290 . . . . . 6 (dom 𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
884, 87syl 17 . . . . 5 (𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
8988adantl 480 . . . 4 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
90893ad2ant2 1131 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
91 fveqeq2 6900 . . . . . . 7 (𝑖 = 𝑗 → ((𝐸𝑖) = {𝑁} ↔ (𝐸𝑗) = {𝑁}))
9291elrab 3675 . . . . . 6 (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ↔ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁}))
93 eldifn 4120 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ {𝑁})
94 eleq2 2814 . . . . . . . . . . . . . . . 16 ((𝐸𝑗) = {𝑁} → (𝑣 ∈ (𝐸𝑗) ↔ 𝑣 ∈ {𝑁}))
9594notbid 317 . . . . . . . . . . . . . . 15 ((𝐸𝑗) = {𝑁} → (¬ 𝑣 ∈ (𝐸𝑗) ↔ ¬ 𝑣 ∈ {𝑁}))
9693, 95imbitrrid 245 . . . . . . . . . . . . . 14 ((𝐸𝑗) = {𝑁} → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸𝑗)))
9796adantl 480 . . . . . . . . . . . . 13 ((𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁}) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸𝑗)))
9897adantl 480 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸𝑗)))
9998imp 405 . . . . . . . . . . 11 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ 𝑣 ∈ (𝐸𝑗))
10099intnand 487 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗)))
101100intnand 487 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
102101ralrimiva 3136 . . . . . . . 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 4995 . . . . . . . . . 10 (𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
104103notbii 319 . . . . . . . . 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 3062 . . . . . . . . 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 6891 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝐸𝑖) = (𝐸𝑗))
107106eleq2d 2811 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ (𝐸𝑗)))
108106eleq2d 2811 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑣 ∈ (𝐸𝑖) ↔ 𝑣 ∈ (𝐸𝑗)))
109107, 108anbi12d 630 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
110109elrab 3675 . . . . . . . . . . 11 (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
111110notbii 319 . . . . . . . . . 10 𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
112111ralbii 3083 . . . . . . . . 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 298 . . . . . . . 8 𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
114102, 113sylibr 233 . . . . . . 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 411 . . . . . 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, 115biimtrid 241 . . . . 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 3135 . . . 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 4445 . . . 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 233 . . 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 14371 . . 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 1368 . 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 28998 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}) = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
1231223adant2 1128 . . 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 6895 . 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 2771 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 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  wne 2930  wral 3051  wrex 3060  {crab 3419  cdif 3937  cun 3938  cin 3939  c0 4318  {csn 4624  {cpr 4626   ciun 4991  Disj wdisj 5108  dom cdm 5672  Fun wfun 6536  cfv 6542  (class class class)co 7415  Fincfn 8960   + caddc 11139  chash 14319  Σcsu 15662  Vtxcvtx 28851  iEdgciedg 28852  Edgcedg 28902  UHGraphcuhgr 28911  UPGraphcupgr 28935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-inf2 9662  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213  ax-pre-sup 11214
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-disj 5109  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-2o 8484  df-oadd 8487  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-sup 9463  df-oi 9531  df-dju 9922  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-div 11900  df-nn 12241  df-2 12303  df-3 12304  df-n0 12501  df-xnn0 12573  df-z 12587  df-uz 12851  df-rp 13005  df-fz 13515  df-fzo 13658  df-seq 13997  df-exp 14057  df-hash 14320  df-cj 15076  df-re 15077  df-im 15078  df-sqrt 15212  df-abs 15213  df-clim 15462  df-sum 15663  df-edg 28903  df-uhgr 28913  df-upgr 28937
This theorem is referenced by:  finsumvtxdg2ssteplem3  29403
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