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Theorem numedglnl 26612
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 8596 . . . . . . 7 (𝑉 ∈ Fin → (𝑉 ∖ {𝑁}) ∈ Fin)
21adantr 481 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (𝑉 ∖ {𝑁}) ∈ Fin)
323ad2ant2 1127 . . . . 5 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (𝑉 ∖ {𝑁}) ∈ Fin)
4 dmfi 8648 . . . . . . . . 9 (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
5 rabfi 8589 . . . . . . . . 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 482 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
873ad2ant2 1127 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
98adantr 481 . . . . 5 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
10 notnotb 316 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝐸𝑖) ↔ ¬ ¬ 𝑁 ∈ (𝐸𝑖))
11 notnotb 316 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ (𝐸𝑖) ↔ ¬ ¬ 𝑣 ∈ (𝐸𝑖))
12 upgruhgr 26570 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
13 edglnl.e . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐸 = (iEdg‘𝐺)
1413uhgrfun 26534 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺 ∈ UHGraph → Fun 𝐸)
1512, 14syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐺 ∈ UPGraph → Fun 𝐸)
1613iedgedg 26518 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Fun 𝐸𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
1715, 16sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ UPGraph ∧ 𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
18 edglnl.v . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑉 = (Vtx‘𝐺)
19 eqid 2795 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Edg‘𝐺) = (Edg‘𝐺)
2018, 19upgredg 26605 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ UPGraph ∧ (𝐸𝑖) ∈ (Edg‘𝐺)) → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛})
2117, 20syldan 591 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ UPGraph ∧ 𝑖 ∈ dom 𝐸) → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛})
2221ex 413 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 ∈ UPGraph → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
23223ad2ant1 1126 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
2423adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
2524adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁}))) ∧ ¬ 𝑣 = 𝑤) → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
2625imp 407 . . . . . . . . . . . . . . . . . . 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 4629 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣𝑁)
28 eldifsni 4629 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ (𝑉 ∖ {𝑁}) → 𝑤𝑁)
29 3elpr2eq 4744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑁 ∈ {𝑚, 𝑛} ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑤 ∈ {𝑚, 𝑛}) ∧ (𝑣𝑁𝑤𝑁)) → 𝑣 = 𝑤)
3029expcom 414 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑣𝑁𝑤𝑁) → ((𝑁 ∈ {𝑚, 𝑛} ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑤 ∈ {𝑚, 𝑛}) → 𝑣 = 𝑤))
31303expd 1346 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1112 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) → (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛})))
38 eleq2 2871 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ {𝑚, 𝑛}))
39 eleq2 2871 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐸𝑖) = {𝑚, 𝑛} → (𝑣 ∈ (𝐸𝑖) ↔ 𝑣 ∈ {𝑚, 𝑛}))
40 eleq2 2871 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐸𝑖) = {𝑚, 𝑛} → (𝑤 ∈ (𝐸𝑖) ↔ 𝑤 ∈ {𝑚, 𝑛}))
4140notbid 319 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐸𝑖) = {𝑚, 𝑛} → (¬ 𝑤 ∈ (𝐸𝑖) ↔ ¬ 𝑤 ∈ {𝑚, 𝑛}))
4239, 41imbi12d 346 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐸𝑖) = {𝑚, 𝑛} → ((𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)) ↔ (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛})))
4338, 42imbi12d 346 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐸𝑖) = {𝑚, 𝑛} → ((𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))) ↔ (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛}))))
4437, 43syl5ibrcom 248 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) → ((𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))))
4544adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) ∧ (𝑚𝑉𝑛𝑉)) → ((𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))))
4645rexlimdvva 3257 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))))
4746ex 413 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑣𝑁𝑤𝑁) → (¬ 𝑣 = 𝑤 → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))))))
4827, 28, 47syl2an 595 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑣 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑤 ∈ (𝑉 ∖ {𝑁})) → (¬ 𝑣 = 𝑤 → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))))))
4948adantl 482 . . . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . 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 244 . . . . . . . . . . . . . . . 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 858 . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . 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 244 . . . . . . . . . . . . 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 858 . . . . . . . . . . . 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 672 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
6059bicomi 225 . . . . . . . . . . . . . 14 (((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
6160notbii 321 . . . . . . . . . . . . 13 (¬ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ ¬ (𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
62 ianor 976 . . . . . . . . . . . . 13 (¬ (𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (¬ 𝑁 ∈ (𝐸𝑖) ∨ ¬ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
63 ianor 976 . . . . . . . . . . . . . 14 (¬ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖)) ↔ (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖)))
6463orbi2i 907 . . . . . . . . . . . . 13 ((¬ 𝑁 ∈ (𝐸𝑖) ∨ ¬ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (¬ 𝑁 ∈ (𝐸𝑖) ∨ (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖))))
6561, 62, 643bitri 298 . . . . . . . . . . . 12 (¬ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (¬ 𝑁 ∈ (𝐸𝑖) ∨ (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖))))
6658, 65sylibr 235 . . . . . . . . . . 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 3149 . . . . . . . . . 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 4195 . . . . . . . . . . . 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 2800 . . . . . . . . . . 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 4258 . . . . . . . . . . 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 276 . . . . . . . . . 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 235 . . . . . . . . 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 413 . . . . . . . 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 858 . . . . . . 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 3158 . . . . . 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 2865 . . . . . . . . 9 (𝑣 = 𝑤 → (𝑣 ∈ (𝐸𝑖) ↔ 𝑤 ∈ (𝐸𝑖)))
7776anbi2d 628 . . . . . . . 8 (𝑣 = 𝑤 → ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
7877rabbidv 3425 . . . . . . 7 (𝑣 = 𝑤 → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} = {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))})
7978disjor 4944 . . . . . 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 235 . . . . 5 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → Disj 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
813, 9, 80hashiun 15010 . . . 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 2801 . . 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 7031 . 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 3149 . . . 4 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
85 iunfi 8658 . . . 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 584 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
87 rabfi 8589 . . . . . 6 (dom 𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
884, 87syl 17 . . . . 5 (𝐸 ∈ Fin → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
8988adantl 482 . . . 4 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
90893ad2ant2 1127 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
91 fveqeq2 6547 . . . . . . 7 (𝑖 = 𝑗 → ((𝐸𝑖) = {𝑁} ↔ (𝐸𝑗) = {𝑁}))
9291elrab 3618 . . . . . 6 (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ↔ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁}))
93 eldifn 4025 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ {𝑁})
94 eleq2 2871 . . . . . . . . . . . . . . . 16 ((𝐸𝑗) = {𝑁} → (𝑣 ∈ (𝐸𝑗) ↔ 𝑣 ∈ {𝑁}))
9594notbid 319 . . . . . . . . . . . . . . 15 ((𝐸𝑗) = {𝑁} → (¬ 𝑣 ∈ (𝐸𝑗) ↔ ¬ 𝑣 ∈ {𝑁}))
9693, 95syl5ibr 247 . . . . . . . . . . . . . 14 ((𝐸𝑗) = {𝑁} → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸𝑗)))
9796adantl 482 . . . . . . . . . . . . 13 ((𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁}) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸𝑗)))
9897adantl 482 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸𝑗)))
9998imp 407 . . . . . . . . . . 11 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ 𝑣 ∈ (𝐸𝑗))
10099intnand 489 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗)))
101100intnand 489 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
102101ralrimiva 3149 . . . . . . . 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 4829 . . . . . . . . . 10 (𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
104103notbii 321 . . . . . . . . 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 3200 . . . . . . . . 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 6538 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝐸𝑖) = (𝐸𝑗))
107106eleq2d 2868 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ (𝐸𝑗)))
108106eleq2d 2868 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑣 ∈ (𝐸𝑖) ↔ 𝑣 ∈ (𝐸𝑗)))
109107, 108anbi12d 630 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
110109elrab 3618 . . . . . . . . . . 11 (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
111110notbii 321 . . . . . . . . . 10 𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
112111ralbii 3132 . . . . . . . . 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 300 . . . . . . . 8 𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
114102, 113sylibr 235 . . . . . . 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 413 . . . . . 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 243 . . . . 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 3148 . . . 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 4314 . . . 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 235 . . 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 13591 . . 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 1364 . 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 26611 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}) = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
1231223adant2 1124 . . 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 6542 . 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 2837 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 396  wo 842  w3a 1080   = wceq 1522  wcel 2081  wne 2984  wral 3105  wrex 3106  {crab 3109  cdif 3856  cun 3857  cin 3858  c0 4211  {csn 4472  {cpr 4474   ciun 4825  Disj wdisj 4930  dom cdm 5443  Fun wfun 6219  cfv 6225  (class class class)co 7016  Fincfn 8357   + caddc 10386  chash 13540  Σcsu 14876  Vtxcvtx 26464  iEdgciedg 26465  Edgcedg 26515  UHGraphcuhgr 26524  UPGraphcupgr 26548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-disj 4931  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-sup 8752  df-oi 8820  df-dju 9176  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-n0 11746  df-xnn0 11816  df-z 11830  df-uz 12094  df-rp 12240  df-fz 12743  df-fzo 12884  df-seq 13220  df-exp 13280  df-hash 13541  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-clim 14679  df-sum 14877  df-edg 26516  df-uhgr 26526  df-upgr 26550
This theorem is referenced by:  finsumvtxdg2ssteplem3  27012
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