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Theorem numedglnl 26084
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 8233 . . . . . . 7 (𝑉 ∈ Fin → (𝑉 ∖ {𝑁}) ∈ Fin)
21adantr 480 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (𝑉 ∖ {𝑁}) ∈ Fin)
323ad2ant2 1103 . . . . 5 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (𝑉 ∖ {𝑁}) ∈ Fin)
4 dmfi 8285 . . . . . . . . 9 (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
5 rabfi 8226 . . . . . . . . 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 481 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
873ad2ant2 1103 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
98adantr 480 . . . . 5 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
10 notnotb 304 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝐸𝑖) ↔ ¬ ¬ 𝑁 ∈ (𝐸𝑖))
11 notnotb 304 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ (𝐸𝑖) ↔ ¬ ¬ 𝑣 ∈ (𝐸𝑖))
12 upgruhgr 26042 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
13 edglnl.e . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐸 = (iEdg‘𝐺)
1413uhgrfun 26006 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺 ∈ UHGraph → Fun 𝐸)
1512, 14syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐺 ∈ UPGraph → Fun 𝐸)
1613iedgedg 25988 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Fun 𝐸𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
1715, 16sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ UPGraph ∧ 𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
18 edglnl.v . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑉 = (Vtx‘𝐺)
19 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Edg‘𝐺) = (Edg‘𝐺)
2018, 19upgredg 26077 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ UPGraph ∧ (𝐸𝑖) ∈ (Edg‘𝐺)) → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛})
2117, 20syldan 486 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ UPGraph ∧ 𝑖 ∈ dom 𝐸) → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛})
2221ex 449 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 ∈ UPGraph → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
23223ad2ant1 1102 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
2423adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
2524adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
2625imp 444 . . . . . . . . . . . . . . . . . . 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 4353 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣𝑁)
28 eldifsni 4353 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ (𝑉 ∖ {𝑁}) → 𝑤𝑁)
29 3elpr2eq 4467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑁 ∈ {𝑚, 𝑛} ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑤 ∈ {𝑚, 𝑛}) ∧ (𝑣𝑁𝑤𝑁)) → 𝑣 = 𝑤)
3029expcom 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑣𝑁𝑤𝑁) → ((𝑁 ∈ {𝑚, 𝑛} ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑤 ∈ {𝑚, 𝑛}) → 𝑣 = 𝑤))
31303expd 1306 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1275 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑣𝑁𝑤𝑁) ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑁 ∈ {𝑚, 𝑛}) → (𝑤 ∈ {𝑚, 𝑛} → 𝑣 = 𝑤))
3433con3d 148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑣𝑁𝑤𝑁) ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑁 ∈ {𝑚, 𝑛}) → (¬ 𝑣 = 𝑤 → ¬ 𝑤 ∈ {𝑚, 𝑛}))
35343exp 1283 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) → (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛})))
38 eleq2 2719 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ {𝑚, 𝑛}))
39 eleq2 2719 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐸𝑖) = {𝑚, 𝑛} → (𝑣 ∈ (𝐸𝑖) ↔ 𝑣 ∈ {𝑚, 𝑛}))
40 eleq2 2719 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐸𝑖) = {𝑚, 𝑛} → (𝑤 ∈ (𝐸𝑖) ↔ 𝑤 ∈ {𝑚, 𝑛}))
4140notbid 307 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐸𝑖) = {𝑚, 𝑛} → (¬ 𝑤 ∈ (𝐸𝑖) ↔ ¬ 𝑤 ∈ {𝑚, 𝑛}))
4239, 41imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐸𝑖) = {𝑚, 𝑛} → ((𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)) ↔ (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛})))
4338, 42imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐸𝑖) = {𝑚, 𝑛} → ((𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))) ↔ (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛}))))
4437, 43syl5ibrcom 237 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) → ((𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))))
4544adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) ∧ (𝑚𝑉𝑛𝑉)) → ((𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))))
4645rexlimdvva 3067 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))))
4746ex 449 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑣𝑁𝑤𝑁) → (¬ 𝑣 = 𝑤 → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))))))
4827, 28, 47syl2an 493 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁})) → (¬ 𝑣 = 𝑤 → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))))))
4948adantl 481 . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . . . 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 392 . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . 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 392 . . . . . . . . . . . 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 888 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
6059bicomi 214 . . . . . . . . . . . . . 14 (((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
6160notbii 309 . . . . . . . . . . . . 13 (¬ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ ¬ (𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
62 ianor 508 . . . . . . . . . . . . 13 (¬ (𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (¬ 𝑁 ∈ (𝐸𝑖) ∨ ¬ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
63 ianor 508 . . . . . . . . . . . . . 14 (¬ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖)) ↔ (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖)))
6463orbi2i 540 . . . . . . . . . . . . 13 ((¬ 𝑁 ∈ (𝐸𝑖) ∨ ¬ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (¬ 𝑁 ∈ (𝐸𝑖) ∨ (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖))))
6561, 62, 643bitri 286 . . . . . . . . . . . 12 (¬ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (¬ 𝑁 ∈ (𝐸𝑖) ∨ (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖))))
6658, 65sylibr 224 . . . . . . . . . . 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 2995 . . . . . . . . . 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 3932 . . . . . . . . . . . 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 2656 . . . . . . . . . . 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 3990 . . . . . . . . . . 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 264 . . . . . . . . . 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 224 . . . . . . . . 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 449 . . . . . . . 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 392 . . . . . . 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 3000 . . . . . 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 2713 . . . . . . . . 9 (𝑣 = 𝑤 → (𝑣 ∈ (𝐸𝑖) ↔ 𝑤 ∈ (𝐸𝑖)))
7776anbi2d 740 . . . . . . . 8 (𝑣 = 𝑤 → ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
7877rabbidv 3220 . . . . . . 7 (𝑣 = 𝑤 → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} = {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))})
7978disjor 4666 . . . . . 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 224 . . . . 5 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → Disj 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
813, 9, 80hashiun 14598 . . . 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 2657 . . 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 6705 . 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 2995 . . . 4 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
85 iunfi 8295 . . . 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 694 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
87 rabfi 8226 . . . . . 6 (dom 𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
884, 87syl 17 . . . . 5 (𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
8988adantl 481 . . . 4 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
90893ad2ant2 1103 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
91 fveq2 6229 . . . . . . . 8 (𝑖 = 𝑗 → (𝐸𝑖) = (𝐸𝑗))
9291eqeq1d 2653 . . . . . . 7 (𝑖 = 𝑗 → ((𝐸𝑖) = {𝑁} ↔ (𝐸𝑗) = {𝑁}))
9392elrab 3396 . . . . . 6 (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ↔ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁}))
94 eldifn 3766 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ {𝑁})
95 eleq2 2719 . . . . . . . . . . . . . . . 16 ((𝐸𝑗) = {𝑁} → (𝑣 ∈ (𝐸𝑗) ↔ 𝑣 ∈ {𝑁}))
9695notbid 307 . . . . . . . . . . . . . . 15 ((𝐸𝑗) = {𝑁} → (¬ 𝑣 ∈ (𝐸𝑗) ↔ ¬ 𝑣 ∈ {𝑁}))
9794, 96syl5ibr 236 . . . . . . . . . . . . . 14 ((𝐸𝑗) = {𝑁} → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸𝑗)))
9897adantl 481 . . . . . . . . . . . . 13 ((𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁}) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸𝑗)))
9998adantl 481 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸𝑗)))
10099imp 444 . . . . . . . . . . 11 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ 𝑣 ∈ (𝐸𝑗))
101100intnand 982 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗)))
102101intnand 982 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
103102ralrimiva 2995 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) → ∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
104 eliun 4556 . . . . . . . . . 10 (𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
105104notbii 309 . . . . . . . . 9 𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ¬ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
106 ralnex 3021 . . . . . . . . 9 (∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ 𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ¬ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
10791eleq2d 2716 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ (𝐸𝑗)))
10891eleq2d 2716 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑣 ∈ (𝐸𝑖) ↔ 𝑣 ∈ (𝐸𝑗)))
109107, 108anbi12d 747 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
110109elrab 3396 . . . . . . . . . . 11 (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
111110notbii 309 . . . . . . . . . 10 𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
112111ralbii 3009 . . . . . . . . 9 (∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ 𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
113105, 106, 1123bitr2i 288 . . . . . . . 8 𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
114103, 113sylibr 224 . . . . . . 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 449 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → ((𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁}) → ¬ 𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))}))
11693, 115syl5bi 232 . . . . 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 2994 . . . 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 4051 . . . 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 224 . . 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 13209 . . 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 1366 . 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 26083 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}) = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
1231223adant2 1100 . . 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 6233 . 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 2691 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  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  cdif 3604  cun 3605  cin 3606  c0 3948  {csn 4210  {cpr 4212   ciun 4552  Disj wdisj 4652  dom cdm 5143  Fun wfun 5920  cfv 5926  (class class class)co 6690  Fincfn 7997   + caddc 9977  #chash 13157  Σcsu 14460  Vtxcvtx 25919  iEdgciedg 25920  Edgcedg 25984  UHGraphcuhgr 25996  UPGraphcupgr 26020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-edg 25985  df-uhgr 25998  df-upgr 26022
This theorem is referenced by:  finsumvtxdg2ssteplem3  26499
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