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Theorem edglnl 26229
Description: The edges incident with a vertex 𝑁 are the edges joining 𝑁 with other vertices and the loops on 𝑁 in a pseudograph. (Contributed by AV, 18-Dec-2021.)
Hypotheses
Ref Expression
edglnl.v 𝑉 = (Vtx‘𝐺)
edglnl.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
edglnl ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( 𝑣 ∈ (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
Allowed substitution hints:   𝐸(𝑖)   𝐺(𝑣)

Proof of Theorem edglnl
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iunrab 4711 . . . 4 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} = {𝑖 ∈ dom 𝐸 ∣ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))}
21a1i 11 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} = {𝑖 ∈ dom 𝐸 ∣ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))})
32uneq1d 3901 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}) = ({𝑖 ∈ dom 𝐸 ∣ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}))
4 unrab 4033 . . 3 ({𝑖 ∈ dom 𝐸 ∣ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}) = {𝑖 ∈ dom 𝐸 ∣ (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁})}
5 simpl 474 . . . . . . . 8 ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) → 𝑁 ∈ (𝐸𝑖))
65rexlimivw 3159 . . . . . . 7 (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) → 𝑁 ∈ (𝐸𝑖))
76a1i 11 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) → 𝑁 ∈ (𝐸𝑖)))
8 snidg 4343 . . . . . . . 8 (𝑁𝑉𝑁 ∈ {𝑁})
98ad2antlr 765 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → 𝑁 ∈ {𝑁})
10 eleq2 2820 . . . . . . 7 ((𝐸𝑖) = {𝑁} → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ {𝑁}))
119, 10syl5ibrcom 237 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝐸𝑖) = {𝑁} → 𝑁 ∈ (𝐸𝑖)))
127, 11jaod 394 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁}) → 𝑁 ∈ (𝐸𝑖)))
13 upgruhgr 26188 . . . . . . . . 9 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
14 edglnl.e . . . . . . . . . 10 𝐸 = (iEdg‘𝐺)
1514uhgrfun 26152 . . . . . . . . 9 (𝐺 ∈ UHGraph → Fun 𝐸)
1613, 15syl 17 . . . . . . . 8 (𝐺 ∈ UPGraph → Fun 𝐸)
1716adantr 472 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → Fun 𝐸)
1814iedgedg 26134 . . . . . . 7 ((Fun 𝐸𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
1917, 18sylan 489 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
20 edglnl.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
21 eqid 2752 . . . . . . . . . 10 (Edg‘𝐺) = (Edg‘𝐺)
2220, 21upgredg 26223 . . . . . . . . 9 ((𝐺 ∈ UPGraph ∧ (𝐸𝑖) ∈ (Edg‘𝐺)) → ∃𝑛𝑉𝑚𝑉 (𝐸𝑖) = {𝑛, 𝑚})
2322ex 449 . . . . . . . 8 (𝐺 ∈ UPGraph → ((𝐸𝑖) ∈ (Edg‘𝐺) → ∃𝑛𝑉𝑚𝑉 (𝐸𝑖) = {𝑛, 𝑚}))
2423ad2antrr 764 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝐸𝑖) ∈ (Edg‘𝐺) → ∃𝑛𝑉𝑚𝑉 (𝐸𝑖) = {𝑛, 𝑚}))
25 dfsn2 4326 . . . . . . . . . . . . . . . . . . . . . 22 {𝑛} = {𝑛, 𝑛}
2625eqcomi 2761 . . . . . . . . . . . . . . . . . . . . 21 {𝑛, 𝑛} = {𝑛}
27 elsni 4330 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ {𝑛} → 𝑁 = 𝑛)
28 sneq 4323 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 = 𝑛 → {𝑁} = {𝑛})
2928eqcomd 2758 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 = 𝑛 → {𝑛} = {𝑁})
3027, 29syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ {𝑛} → {𝑛} = {𝑁})
3126, 30syl5eq 2798 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ {𝑛} → {𝑛, 𝑛} = {𝑁})
3231, 26eleq2s 2849 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ {𝑛, 𝑛} → {𝑛, 𝑛} = {𝑁})
33 preq2 4405 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → {𝑛, 𝑚} = {𝑛, 𝑛})
3433eleq2d 2817 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → (𝑁 ∈ {𝑛, 𝑚} ↔ 𝑁 ∈ {𝑛, 𝑛}))
3533eqeq1d 2754 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → ({𝑛, 𝑚} = {𝑁} ↔ {𝑛, 𝑛} = {𝑁}))
3634, 35imbi12d 333 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑛 → ((𝑁 ∈ {𝑛, 𝑚} → {𝑛, 𝑚} = {𝑁}) ↔ (𝑁 ∈ {𝑛, 𝑛} → {𝑛, 𝑛} = {𝑁})))
3732, 36mpbiri 248 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑛 → (𝑁 ∈ {𝑛, 𝑚} → {𝑛, 𝑚} = {𝑁}))
3837imp 444 . . . . . . . . . . . . . . . . 17 ((𝑚 = 𝑛𝑁 ∈ {𝑛, 𝑚}) → {𝑛, 𝑚} = {𝑁})
3938olcd 407 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑛𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))
4039expcom 450 . . . . . . . . . . . . . . 15 (𝑁 ∈ {𝑛, 𝑚} → (𝑚 = 𝑛 → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
41403ad2ant3 1129 . . . . . . . . . . . . . 14 ((𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚}) → (𝑚 = 𝑛 → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
4241com12 32 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
43 simpr3 1235 . . . . . . . . . . . . . . . 16 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑁 ∈ {𝑛, 𝑚})
44 simpl 474 . . . . . . . . . . . . . . . . . 18 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑚𝑛)
4544necomd 2979 . . . . . . . . . . . . . . . . 17 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑛𝑚)
46 simpr2 1233 . . . . . . . . . . . . . . . . 17 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → (𝑛𝑉𝑚𝑉))
47 prproe 4578 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ {𝑛, 𝑚} ∧ 𝑛𝑚 ∧ (𝑛𝑉𝑚𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚})
4843, 45, 46, 47syl3anc 1473 . . . . . . . . . . . . . . . 16 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚})
49 r19.42v 3222 . . . . . . . . . . . . . . . 16 (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ↔ (𝑁 ∈ {𝑛, 𝑚} ∧ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚}))
5043, 48, 49sylanbrc 701 . . . . . . . . . . . . . . 15 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}))
5150orcd 406 . . . . . . . . . . . . . 14 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))
5251ex 449 . . . . . . . . . . . . 13 (𝑚𝑛 → ((𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
5342, 52pm2.61ine 3007 . . . . . . . . . . . 12 ((𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))
54533exp 1112 . . . . . . . . . . 11 (𝑁𝑉 → ((𝑛𝑉𝑚𝑉) → (𝑁 ∈ {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))))
5554ad2antlr 765 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝑛𝑉𝑚𝑉) → (𝑁 ∈ {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))))
5655imp 444 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) ∧ (𝑛𝑉𝑚𝑉)) → (𝑁 ∈ {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
57 eleq2 2820 . . . . . . . . . 10 ((𝐸𝑖) = {𝑛, 𝑚} → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ {𝑛, 𝑚}))
58 eleq2 2820 . . . . . . . . . . . . 13 ((𝐸𝑖) = {𝑛, 𝑚} → (𝑣 ∈ (𝐸𝑖) ↔ 𝑣 ∈ {𝑛, 𝑚}))
5957, 58anbi12d 749 . . . . . . . . . . . 12 ((𝐸𝑖) = {𝑛, 𝑚} → ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ (𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚})))
6059rexbidv 3182 . . . . . . . . . . 11 ((𝐸𝑖) = {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚})))
61 eqeq1 2756 . . . . . . . . . . 11 ((𝐸𝑖) = {𝑛, 𝑚} → ((𝐸𝑖) = {𝑁} ↔ {𝑛, 𝑚} = {𝑁}))
6260, 61orbi12d 748 . . . . . . . . . 10 ((𝐸𝑖) = {𝑛, 𝑚} → ((∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁}) ↔ (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
6357, 62imbi12d 333 . . . . . . . . 9 ((𝐸𝑖) = {𝑛, 𝑚} → ((𝑁 ∈ (𝐸𝑖) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁})) ↔ (𝑁 ∈ {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))))
6456, 63syl5ibrcom 237 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) ∧ (𝑛𝑉𝑚𝑉)) → ((𝐸𝑖) = {𝑛, 𝑚} → (𝑁 ∈ (𝐸𝑖) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁}))))
6564rexlimdvva 3168 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → (∃𝑛𝑉𝑚𝑉 (𝐸𝑖) = {𝑛, 𝑚} → (𝑁 ∈ (𝐸𝑖) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁}))))
6624, 65syld 47 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝐸𝑖) ∈ (Edg‘𝐺) → (𝑁 ∈ (𝐸𝑖) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁}))))
6719, 66mpd 15 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → (𝑁 ∈ (𝐸𝑖) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁})))
6812, 67impbid 202 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁}) ↔ 𝑁 ∈ (𝐸𝑖)))
6968rabbidva 3320 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → {𝑖 ∈ dom 𝐸 ∣ (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁})} = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
704, 69syl5eq 2798 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ({𝑖 ∈ dom 𝐸 ∣ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}) = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
713, 70eqtrd 2786 1 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → ( 𝑣 ∈ (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:  wi 4  wo 382  wa 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wrex 3043  {crab 3046  cdif 3704  cun 3705  {csn 4313  {cpr 4315   ciun 4664  dom cdm 5258  Fun wfun 6035  cfv 6041  Vtxcvtx 26065  iEdgciedg 26066  Edgcedg 26130  UHGraphcuhgr 26142  UPGraphcupgr 26166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-oadd 7725  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8947  df-cda 9174  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-n0 11477  df-xnn0 11548  df-z 11562  df-uz 11872  df-fz 12512  df-hash 13304  df-edg 26131  df-uhgr 26144  df-upgr 26168
This theorem is referenced by:  numedglnl  26230
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