MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  numedglnl Structured version   Visualization version   GIF version

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) ∧ 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 9215 . . . . . . 7 (𝑉 ∈ Fin → (𝑉 ∖ {𝑁}) ∈ Fin)
21adantr 480 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) → (𝑉 ∖ {𝑁}) ∈ Fin)
323ad2ant2 1135 . . . . 5 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → (𝑉 ∖ {𝑁}) ∈ Fin)
4 dmfi 9375 . . . . . . . . 9 (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
5 rabfi 9303 . . . . . . . . 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 1135 . . . . . 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 315 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝐸𝑖) ↔ ¬ ¬ 𝑁 ∈ (𝐸𝑖))
11 notnotb 315 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ (𝐸𝑖) ↔ ¬ ¬ 𝑣 ∈ (𝐸𝑖))
12 upgruhgr 29119 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
13 edglnl.e . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐸 = (iEdg‘𝐺)
1413uhgrfun 29083 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺 ∈ UHGraph → Fun 𝐸)
1512, 14syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐺 ∈ UPGraph → Fun 𝐸)
1613iedgedg 29067 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Edg‘𝐺) = (Edg‘𝐺)
2018, 19upgredg 29154 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ UPGraph ∧ (𝐸𝑖) ∈ (Edg‘𝐺)) → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛})
2117, 20syldan 591 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐺 ∈ UPGraph ∧ 𝑖 ∈ dom 𝐸) → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛})
2221ex 412 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 ∈ UPGraph → (𝑖 ∈ dom 𝐸 → ∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛}))
23223ad2ant1 1134 . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . 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 4790 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 ∈ (𝑉 ∖ {𝑁}) → 𝑣𝑁)
28 eldifsni 4790 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ (𝑉 ∖ {𝑁}) → 𝑤𝑁)
29 3elpr2eq 4906 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑁 ∈ {𝑚, 𝑛} ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑤 ∈ {𝑚, 𝑛}) ∧ (𝑣𝑁𝑤𝑁)) → 𝑣 = 𝑤)
3029expcom 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑣𝑁𝑤𝑁) → ((𝑁 ∈ {𝑚, 𝑛} ∧ 𝑣 ∈ {𝑚, 𝑛} ∧ 𝑤 ∈ {𝑚, 𝑛}) → 𝑣 = 𝑤))
31303expd 1354 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) → (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛})))
38 eleq2 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ {𝑚, 𝑛}))
39 eleq2 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐸𝑖) = {𝑚, 𝑛} → (𝑣 ∈ (𝐸𝑖) ↔ 𝑣 ∈ {𝑚, 𝑛}))
40 eleq2 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐸𝑖) = {𝑚, 𝑛} → (𝑤 ∈ (𝐸𝑖) ↔ 𝑤 ∈ {𝑚, 𝑛}))
4140notbid 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐸𝑖) = {𝑚, 𝑛} → (¬ 𝑤 ∈ (𝐸𝑖) ↔ ¬ 𝑤 ∈ {𝑚, 𝑛}))
4239, 41imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐸𝑖) = {𝑚, 𝑛} → ((𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)) ↔ (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛})))
4338, 42imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐸𝑖) = {𝑚, 𝑛} → ((𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))) ↔ (𝑁 ∈ {𝑚, 𝑛} → (𝑣 ∈ {𝑚, 𝑛} → ¬ 𝑤 ∈ {𝑚, 𝑛}))))
4437, 43syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3213 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑣𝑁𝑤𝑁) ∧ ¬ 𝑣 = 𝑤) → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖)))))
4746ex 412 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑣𝑁𝑤𝑁) → (¬ 𝑣 = 𝑤 → (∃𝑚𝑉𝑛𝑉 (𝐸𝑖) = {𝑚, 𝑛} → (𝑁 ∈ (𝐸𝑖) → (𝑣 ∈ (𝐸𝑖) → ¬ 𝑤 ∈ (𝐸𝑖))))))
4827, 28, 47syl2an 596 . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . . . 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 864 . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . 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 864 . . . . . . . . . . . 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 224 . . . . . . . . . . . . . 14 (((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
6160notbii 320 . . . . . . . . . . . . 13 (¬ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ ¬ (𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
62 ianor 984 . . . . . . . . . . . . 13 (¬ (𝑁 ∈ (𝐸𝑖) ∧ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (¬ 𝑁 ∈ (𝐸𝑖) ∨ ¬ (𝑣 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
63 ianor 984 . . . . . . . . . . . . . 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 297 . . . . . . . . . . . 12 (¬ ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∧ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))) ↔ (¬ 𝑁 ∈ (𝐸𝑖) ∨ (¬ 𝑣 ∈ (𝐸𝑖) ∨ ¬ 𝑤 ∈ (𝐸𝑖))))
6658, 65sylibr 234 . . . . . . . . . . 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 3146 . . . . . . . . . 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 4316 . . . . . . . . . . . 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 4388 . . . . . . . . . . 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 275 . . . . . . . . . 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 234 . . . . . . . . 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 412 . . . . . . . 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 864 . . . . . . 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 3202 . . . . . 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 2824 . . . . . . . . 9 (𝑣 = 𝑤 → (𝑣 ∈ (𝐸𝑖) ↔ 𝑤 ∈ (𝐸𝑖)))
7776anbi2d 630 . . . . . . . 8 (𝑣 = 𝑤 → ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))))
7877rabbidv 3444 . . . . . . 7 (𝑣 = 𝑤 → {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} = {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑤 ∈ (𝐸𝑖))})
7978disjor 5125 . . . . . 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 234 . . . . 5 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → Disj 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
813, 9, 80hashiun 15858 . . . 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 7446 . 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 3146 . . . 4 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → ∀𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∈ Fin)
85 iunfi 9383 . . . 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 9303 . . . . . 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 1135 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) → {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ∈ Fin)
91 fveqeq2 6915 . . . . . . 7 (𝑖 = 𝑗 → ((𝐸𝑖) = {𝑁} ↔ (𝐸𝑗) = {𝑁}))
9291elrab 3692 . . . . . 6 (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}} ↔ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁}))
93 eldifn 4132 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ {𝑁})
94 eleq2 2830 . . . . . . . . . . . . . . . 16 ((𝐸𝑗) = {𝑁} → (𝑣 ∈ (𝐸𝑗) ↔ 𝑣 ∈ {𝑁}))
9594notbid 318 . . . . . . . . . . . . . . 15 ((𝐸𝑗) = {𝑁} → (¬ 𝑣 ∈ (𝐸𝑗) ↔ ¬ 𝑣 ∈ {𝑁}))
9693, 95imbitrrid 246 . . . . . . . . . . . . . 14 ((𝐸𝑗) = {𝑁} → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸𝑗)))
9796adantl 481 . . . . . . . . . . . . 13 ((𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁}) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸𝑗)))
9897adantl 481 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) → (𝑣 ∈ (𝑉 ∖ {𝑁}) → ¬ 𝑣 ∈ (𝐸𝑗)))
9998imp 406 . . . . . . . . . . 11 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ 𝑣 ∈ (𝐸𝑗))
10099intnand 488 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗)))
101100intnand 488 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁𝑉) ∧ (𝑗 ∈ dom 𝐸 ∧ (𝐸𝑗) = {𝑁})) ∧ 𝑣 ∈ (𝑉 ∖ {𝑁})) → ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
102101ralrimiva 3146 . . . . . . . 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 320 . . . . . . . . 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 3072 . . . . . . . . 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 6906 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝐸𝑖) = (𝐸𝑗))
107106eleq2d 2827 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ (𝐸𝑗)))
108106eleq2d 2827 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑣 ∈ (𝐸𝑖) ↔ 𝑣 ∈ (𝐸𝑗)))
109107, 108anbi12d 632 . . . . . . . . . . . 12 (𝑖 = 𝑗 → ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
110109elrab 3692 . . . . . . . . . . 11 (𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
111110notbii 320 . . . . . . . . . 10 𝑗 ∈ {𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
112111ralbii 3093 . . . . . . . . 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 299 . . . . . . . 8 𝑗 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ↔ ∀𝑣 ∈ (𝑉 ∖ {𝑁}) ¬ (𝑗 ∈ dom 𝐸 ∧ (𝑁 ∈ (𝐸𝑗) ∧ 𝑣 ∈ (𝐸𝑗))))
114102, 113sylibr 234 . . . . . . 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 412 . . . . . 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 242 . . . . 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 3145 . . . 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 4451 . . . 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 234 . . 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 14421 . . 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 29160 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}) = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
1231223adant2 1132 . . 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 6910 . 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 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
  Copyright terms: Public domain W3C validator