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Theorem edglnl 27035
 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 4941 . . . 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 4067 . 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 4208 . . 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 486 . . . . . . . 8 ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) → 𝑁 ∈ (𝐸𝑖))
65rexlimivw 3206 . . . . . . 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 4556 . . . . . . . 8 (𝑁𝑉𝑁 ∈ {𝑁})
98ad2antlr 726 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → 𝑁 ∈ {𝑁})
10 eleq2 2840 . . . . . . 7 ((𝐸𝑖) = {𝑁} → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ {𝑁}))
119, 10syl5ibrcom 250 . . . . . 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 26994 . . . . . . . . 9 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
14 edglnl.e . . . . . . . . . 10 𝐸 = (iEdg‘𝐺)
1514uhgrfun 26958 . . . . . . . . 9 (𝐺 ∈ UHGraph → Fun 𝐸)
1613, 15syl 17 . . . . . . . 8 (𝐺 ∈ UPGraph → Fun 𝐸)
1716adantr 484 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → Fun 𝐸)
1814iedgedg 26942 . . . . . . 7 ((Fun 𝐸𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
1917, 18sylan 583 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
20 edglnl.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
21 eqid 2758 . . . . . . . . . 10 (Edg‘𝐺) = (Edg‘𝐺)
2220, 21upgredg 27029 . . . . . . . . 9 ((𝐺 ∈ UPGraph ∧ (𝐸𝑖) ∈ (Edg‘𝐺)) → ∃𝑛𝑉𝑚𝑉 (𝐸𝑖) = {𝑛, 𝑚})
2322ex 416 . . . . . . . 8 (𝐺 ∈ UPGraph → ((𝐸𝑖) ∈ (Edg‘𝐺) → ∃𝑛𝑉𝑚𝑉 (𝐸𝑖) = {𝑛, 𝑚}))
2423ad2antrr 725 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝐸𝑖) ∈ (Edg‘𝐺) → ∃𝑛𝑉𝑚𝑉 (𝐸𝑖) = {𝑛, 𝑚}))
25 dfsn2 4535 . . . . . . . . . . . . . . . . . . . . . 22 {𝑛} = {𝑛, 𝑛}
2625eqcomi 2767 . . . . . . . . . . . . . . . . . . . . 21 {𝑛, 𝑛} = {𝑛}
27 elsni 4539 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ {𝑛} → 𝑁 = 𝑛)
28 sneq 4532 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 = 𝑛 → {𝑁} = {𝑛})
2928eqcomd 2764 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 = 𝑛 → {𝑛} = {𝑁})
3027, 29syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ {𝑛} → {𝑛} = {𝑁})
3126, 30syl5eq 2805 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ {𝑛} → {𝑛, 𝑛} = {𝑁})
3231, 26eleq2s 2870 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ {𝑛, 𝑛} → {𝑛, 𝑛} = {𝑁})
33 preq2 4627 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → {𝑛, 𝑚} = {𝑛, 𝑛})
3433eleq2d 2837 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → (𝑁 ∈ {𝑛, 𝑚} ↔ 𝑁 ∈ {𝑛, 𝑛}))
3533eqeq1d 2760 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → ({𝑛, 𝑚} = {𝑁} ↔ {𝑛, 𝑛} = {𝑁}))
3634, 35imbi12d 348 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑛 → ((𝑁 ∈ {𝑛, 𝑚} → {𝑛, 𝑚} = {𝑁}) ↔ (𝑁 ∈ {𝑛, 𝑛} → {𝑛, 𝑛} = {𝑁})))
3732, 36mpbiri 261 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑛 → (𝑁 ∈ {𝑛, 𝑚} → {𝑛, 𝑚} = {𝑁}))
3837imp 410 . . . . . . . . . . . . . . . . 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 417 . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . . . 18 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑚𝑛)
4544necomd 3006 . . . . . . . . . . . . . . . . 17 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑛𝑚)
46 simpr2 1192 . . . . . . . . . . . . . . . . 17 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → (𝑛𝑉𝑚𝑉))
47 prproe 4796 . . . . . . . . . . . . . . . . 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 3268 . . . . . . . . . . . . . . . 16 (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ↔ (𝑁 ∈ {𝑛, 𝑚} ∧ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚}))
5043, 48, 49sylanbrc 586 . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . 13 (𝑚𝑛 → ((𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
5342, 52pm2.61ine 3034 . . . . . . . . . . . 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 726 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝑛𝑉𝑚𝑉) → (𝑁 ∈ {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))))
5655imp 410 . . . . . . . . 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 2840 . . . . . . . . . 10 ((𝐸𝑖) = {𝑛, 𝑚} → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ {𝑛, 𝑚}))
58 eleq2 2840 . . . . . . . . . . . . 13 ((𝐸𝑖) = {𝑛, 𝑚} → (𝑣 ∈ (𝐸𝑖) ↔ 𝑣 ∈ {𝑛, 𝑚}))
5957, 58anbi12d 633 . . . . . . . . . . . 12 ((𝐸𝑖) = {𝑛, 𝑚} → ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ (𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚})))
6059rexbidv 3221 . . . . . . . . . . 11 ((𝐸𝑖) = {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚})))
61 eqeq1 2762 . . . . . . . . . . 11 ((𝐸𝑖) = {𝑛, 𝑚} → ((𝐸𝑖) = {𝑁} ↔ {𝑛, 𝑚} = {𝑁}))
6260, 61orbi12d 916 . . . . . . . . . 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 348 . . . . . . . . 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 250 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) ∧ (𝑛𝑉𝑚𝑉)) → ((𝐸𝑖) = {𝑛, 𝑚} → (𝑁 ∈ (𝐸𝑖) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁}))))
6564rexlimdvva 3218 . . . . . . 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 215 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁}) ↔ 𝑁 ∈ (𝐸𝑖)))
6968rabbidva 3390 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → {𝑖 ∈ dom 𝐸 ∣ (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁})} = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
704, 69syl5eq 2805 . 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 2793 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 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∃wrex 3071  {crab 3074   ∖ cdif 3855   ∪ cun 3856  {csn 4522  {cpr 4524  ∪ ciun 4883  dom cdm 5524  Fun wfun 6329  ‘cfv 6335  Vtxcvtx 26888  iEdgciedg 26889  Edgcedg 26939  UHGraphcuhgr 26948  UPGraphcupgr 26972 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-2o 8113  df-oadd 8116  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-dju 9363  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-n0 11935  df-xnn0 12007  df-z 12021  df-uz 12283  df-fz 12940  df-hash 13741  df-edg 26940  df-uhgr 26950  df-upgr 26974 This theorem is referenced by:  numedglnl  27036
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