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Theorem edglnl 28907
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 5048 . . . 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 4157 . 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 4300 . . 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 482 . . . . . . . 8 ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) → 𝑁 ∈ (𝐸𝑖))
65rexlimivw 3145 . . . . . . 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 4657 . . . . . . . 8 (𝑁𝑉𝑁 ∈ {𝑁})
98ad2antlr 724 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → 𝑁 ∈ {𝑁})
10 eleq2 2816 . . . . . . 7 ((𝐸𝑖) = {𝑁} → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ {𝑁}))
119, 10syl5ibrcom 246 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝐸𝑖) = {𝑁} → 𝑁 ∈ (𝐸𝑖)))
127, 11jaod 856 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁}) → 𝑁 ∈ (𝐸𝑖)))
13 upgruhgr 28866 . . . . . . . . 9 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
14 edglnl.e . . . . . . . . . 10 𝐸 = (iEdg‘𝐺)
1514uhgrfun 28830 . . . . . . . . 9 (𝐺 ∈ UHGraph → Fun 𝐸)
1613, 15syl 17 . . . . . . . 8 (𝐺 ∈ UPGraph → Fun 𝐸)
1716adantr 480 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → Fun 𝐸)
1814iedgedg 28814 . . . . . . 7 ((Fun 𝐸𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
1917, 18sylan 579 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
20 edglnl.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
21 eqid 2726 . . . . . . . . . 10 (Edg‘𝐺) = (Edg‘𝐺)
2220, 21upgredg 28901 . . . . . . . . 9 ((𝐺 ∈ UPGraph ∧ (𝐸𝑖) ∈ (Edg‘𝐺)) → ∃𝑛𝑉𝑚𝑉 (𝐸𝑖) = {𝑛, 𝑚})
2322ex 412 . . . . . . . 8 (𝐺 ∈ UPGraph → ((𝐸𝑖) ∈ (Edg‘𝐺) → ∃𝑛𝑉𝑚𝑉 (𝐸𝑖) = {𝑛, 𝑚}))
2423ad2antrr 723 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝐸𝑖) ∈ (Edg‘𝐺) → ∃𝑛𝑉𝑚𝑉 (𝐸𝑖) = {𝑛, 𝑚}))
25 dfsn2 4636 . . . . . . . . . . . . . . . . . . . . . 22 {𝑛} = {𝑛, 𝑛}
2625eqcomi 2735 . . . . . . . . . . . . . . . . . . . . 21 {𝑛, 𝑛} = {𝑛}
27 elsni 4640 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ {𝑛} → 𝑁 = 𝑛)
28 sneq 4633 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 = 𝑛 → {𝑁} = {𝑛})
2928eqcomd 2732 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 = 𝑛 → {𝑛} = {𝑁})
3027, 29syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ {𝑛} → {𝑛} = {𝑁})
3126, 30eqtrid 2778 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ {𝑛} → {𝑛, 𝑛} = {𝑁})
3231, 26eleq2s 2845 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ {𝑛, 𝑛} → {𝑛, 𝑛} = {𝑁})
33 preq2 4733 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → {𝑛, 𝑚} = {𝑛, 𝑛})
3433eleq2d 2813 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → (𝑁 ∈ {𝑛, 𝑚} ↔ 𝑁 ∈ {𝑛, 𝑛}))
3533eqeq1d 2728 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → ({𝑛, 𝑚} = {𝑁} ↔ {𝑛, 𝑛} = {𝑁}))
3634, 35imbi12d 344 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑛 → ((𝑁 ∈ {𝑛, 𝑚} → {𝑛, 𝑚} = {𝑁}) ↔ (𝑁 ∈ {𝑛, 𝑛} → {𝑛, 𝑛} = {𝑁})))
3732, 36mpbiri 258 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑛 → (𝑁 ∈ {𝑛, 𝑚} → {𝑛, 𝑚} = {𝑁}))
3837imp 406 . . . . . . . . . . . . . . . . 17 ((𝑚 = 𝑛𝑁 ∈ {𝑛, 𝑚}) → {𝑛, 𝑚} = {𝑁})
3938olcd 871 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑛𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))
4039expcom 413 . . . . . . . . . . . . . . 15 (𝑁 ∈ {𝑛, 𝑚} → (𝑚 = 𝑛 → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
41403ad2ant3 1132 . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . . 16 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑁 ∈ {𝑛, 𝑚})
44 simpl 482 . . . . . . . . . . . . . . . . . 18 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑚𝑛)
4544necomd 2990 . . . . . . . . . . . . . . . . 17 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑛𝑚)
46 simpr2 1192 . . . . . . . . . . . . . . . . 17 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → (𝑛𝑉𝑚𝑉))
47 prproe 4900 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ {𝑛, 𝑚} ∧ 𝑛𝑚 ∧ (𝑛𝑉𝑚𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚})
4843, 45, 46, 47syl3anc 1368 . . . . . . . . . . . . . . . 16 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚})
49 r19.42v 3184 . . . . . . . . . . . . . . . 16 (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ↔ (𝑁 ∈ {𝑛, 𝑚} ∧ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚}))
5043, 48, 49sylanbrc 582 . . . . . . . . . . . . . . 15 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}))
5150orcd 870 . . . . . . . . . . . . . 14 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))
5251ex 412 . . . . . . . . . . . . 13 (𝑚𝑛 → ((𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
5342, 52pm2.61ine 3019 . . . . . . . . . . . 12 ((𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))
54533exp 1116 . . . . . . . . . . 11 (𝑁𝑉 → ((𝑛𝑉𝑚𝑉) → (𝑁 ∈ {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))))
5554ad2antlr 724 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝑛𝑉𝑚𝑉) → (𝑁 ∈ {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))))
5655imp 406 . . . . . . . . 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 2816 . . . . . . . . . 10 ((𝐸𝑖) = {𝑛, 𝑚} → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ {𝑛, 𝑚}))
58 eleq2 2816 . . . . . . . . . . . . 13 ((𝐸𝑖) = {𝑛, 𝑚} → (𝑣 ∈ (𝐸𝑖) ↔ 𝑣 ∈ {𝑛, 𝑚}))
5957, 58anbi12d 630 . . . . . . . . . . . 12 ((𝐸𝑖) = {𝑛, 𝑚} → ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ (𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚})))
6059rexbidv 3172 . . . . . . . . . . 11 ((𝐸𝑖) = {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚})))
61 eqeq1 2730 . . . . . . . . . . 11 ((𝐸𝑖) = {𝑛, 𝑚} → ((𝐸𝑖) = {𝑁} ↔ {𝑛, 𝑚} = {𝑁}))
6260, 61orbi12d 915 . . . . . . . . . 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 344 . . . . . . . . 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 246 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) ∧ (𝑛𝑉𝑚𝑉)) → ((𝐸𝑖) = {𝑛, 𝑚} → (𝑁 ∈ (𝐸𝑖) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁}))))
6564rexlimdvva 3205 . . . . . . 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 211 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁}) ↔ 𝑁 ∈ (𝐸𝑖)))
6968rabbidva 3433 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → {𝑖 ∈ dom 𝐸 ∣ (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁})} = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
704, 69eqtrid 2778 . 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 2766 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  wa 395  wo 844  w3a 1084   = wceq 1533  wcel 2098  wne 2934  wrex 3064  {crab 3426  cdif 3940  cun 3941  {csn 4623  {cpr 4625   ciun 4990  dom cdm 5669  Fun wfun 6530  cfv 6536  Vtxcvtx 28760  iEdgciedg 28761  Edgcedg 28811  UHGraphcuhgr 28820  UPGraphcupgr 28844
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-2o 8465  df-oadd 8468  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-dju 9895  df-card 9933  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-n0 12474  df-xnn0 12546  df-z 12560  df-uz 12824  df-fz 13488  df-hash 14294  df-edg 28812  df-uhgr 28822  df-upgr 28846
This theorem is referenced by:  numedglnl  28908
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