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Theorem edglnl 28094
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 5012 . . . 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 4122 . 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 4265 . . 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 483 . . . . . . . 8 ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) → 𝑁 ∈ (𝐸𝑖))
65rexlimivw 3148 . . . . . . 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 4620 . . . . . . . 8 (𝑁𝑉𝑁 ∈ {𝑁})
98ad2antlr 725 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → 𝑁 ∈ {𝑁})
10 eleq2 2826 . . . . . . 7 ((𝐸𝑖) = {𝑁} → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ {𝑁}))
119, 10syl5ibrcom 246 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝐸𝑖) = {𝑁} → 𝑁 ∈ (𝐸𝑖)))
127, 11jaod 857 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁}) → 𝑁 ∈ (𝐸𝑖)))
13 upgruhgr 28053 . . . . . . . . 9 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
14 edglnl.e . . . . . . . . . 10 𝐸 = (iEdg‘𝐺)
1514uhgrfun 28017 . . . . . . . . 9 (𝐺 ∈ UHGraph → Fun 𝐸)
1613, 15syl 17 . . . . . . . 8 (𝐺 ∈ UPGraph → Fun 𝐸)
1716adantr 481 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → Fun 𝐸)
1814iedgedg 28001 . . . . . . 7 ((Fun 𝐸𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
1917, 18sylan 580 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
20 edglnl.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
21 eqid 2736 . . . . . . . . . 10 (Edg‘𝐺) = (Edg‘𝐺)
2220, 21upgredg 28088 . . . . . . . . 9 ((𝐺 ∈ UPGraph ∧ (𝐸𝑖) ∈ (Edg‘𝐺)) → ∃𝑛𝑉𝑚𝑉 (𝐸𝑖) = {𝑛, 𝑚})
2322ex 413 . . . . . . . 8 (𝐺 ∈ UPGraph → ((𝐸𝑖) ∈ (Edg‘𝐺) → ∃𝑛𝑉𝑚𝑉 (𝐸𝑖) = {𝑛, 𝑚}))
2423ad2antrr 724 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝐸𝑖) ∈ (Edg‘𝐺) → ∃𝑛𝑉𝑚𝑉 (𝐸𝑖) = {𝑛, 𝑚}))
25 dfsn2 4599 . . . . . . . . . . . . . . . . . . . . . 22 {𝑛} = {𝑛, 𝑛}
2625eqcomi 2745 . . . . . . . . . . . . . . . . . . . . 21 {𝑛, 𝑛} = {𝑛}
27 elsni 4603 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ {𝑛} → 𝑁 = 𝑛)
28 sneq 4596 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 = 𝑛 → {𝑁} = {𝑛})
2928eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 = 𝑛 → {𝑛} = {𝑁})
3027, 29syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ {𝑛} → {𝑛} = {𝑁})
3126, 30eqtrid 2788 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ {𝑛} → {𝑛, 𝑛} = {𝑁})
3231, 26eleq2s 2856 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ {𝑛, 𝑛} → {𝑛, 𝑛} = {𝑁})
33 preq2 4695 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → {𝑛, 𝑚} = {𝑛, 𝑛})
3433eleq2d 2823 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → (𝑁 ∈ {𝑛, 𝑚} ↔ 𝑁 ∈ {𝑛, 𝑛}))
3533eqeq1d 2738 . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . 18 (𝑚 = 𝑛 → (𝑁 ∈ {𝑛, 𝑚} → {𝑛, 𝑚} = {𝑁}))
3837imp 407 . . . . . . . . . . . . . . . . 17 ((𝑚 = 𝑛𝑁 ∈ {𝑛, 𝑚}) → {𝑛, 𝑚} = {𝑁})
3938olcd 872 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝑛𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))
4039expcom 414 . . . . . . . . . . . . . . 15 (𝑁 ∈ {𝑛, 𝑚} → (𝑚 = 𝑛 → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
41403ad2ant3 1135 . . . . . . . . . . . . . 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 1196 . . . . . . . . . . . . . . . 16 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑁 ∈ {𝑛, 𝑚})
44 simpl 483 . . . . . . . . . . . . . . . . . 18 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑚𝑛)
4544necomd 2999 . . . . . . . . . . . . . . . . 17 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑛𝑚)
46 simpr2 1195 . . . . . . . . . . . . . . . . 17 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → (𝑛𝑉𝑚𝑉))
47 prproe 4863 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ {𝑛, 𝑚} ∧ 𝑛𝑚 ∧ (𝑛𝑉𝑚𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚})
4843, 45, 46, 47syl3anc 1371 . . . . . . . . . . . . . . . 16 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚})
49 r19.42v 3187 . . . . . . . . . . . . . . . 16 (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ↔ (𝑁 ∈ {𝑛, 𝑚} ∧ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚}))
5043, 48, 49sylanbrc 583 . . . . . . . . . . . . . . 15 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}))
5150orcd 871 . . . . . . . . . . . . . 14 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))
5251ex 413 . . . . . . . . . . . . 13 (𝑚𝑛 → ((𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
5342, 52pm2.61ine 3028 . . . . . . . . . . . 12 ((𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))
54533exp 1119 . . . . . . . . . . 11 (𝑁𝑉 → ((𝑛𝑉𝑚𝑉) → (𝑁 ∈ {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))))
5554ad2antlr 725 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → ((𝑛𝑉𝑚𝑉) → (𝑁 ∈ {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁}))))
5655imp 407 . . . . . . . . 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 2826 . . . . . . . . . 10 ((𝐸𝑖) = {𝑛, 𝑚} → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ {𝑛, 𝑚}))
58 eleq2 2826 . . . . . . . . . . . . 13 ((𝐸𝑖) = {𝑛, 𝑚} → (𝑣 ∈ (𝐸𝑖) ↔ 𝑣 ∈ {𝑛, 𝑚}))
5957, 58anbi12d 631 . . . . . . . . . . . 12 ((𝐸𝑖) = {𝑛, 𝑚} → ((𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ (𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚})))
6059rexbidv 3175 . . . . . . . . . . 11 ((𝐸𝑖) = {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚})))
61 eqeq1 2740 . . . . . . . . . . 11 ((𝐸𝑖) = {𝑛, 𝑚} → ((𝐸𝑖) = {𝑁} ↔ {𝑛, 𝑚} = {𝑁}))
6260, 61orbi12d 917 . . . . . . . . . 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 3414 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → {𝑖 ∈ dom 𝐸 ∣ (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁})} = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
704, 69eqtrid 2788 . 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 2776 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 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wrex 3073  {crab 3407  cdif 3907  cun 3908  {csn 4586  {cpr 4588   ciun 4954  dom cdm 5633  Fun wfun 6490  cfv 6496  Vtxcvtx 27947  iEdgciedg 27948  Edgcedg 27998  UHGraphcuhgr 28007  UPGraphcupgr 28031
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-edg 27999  df-uhgr 28009  df-upgr 28033
This theorem is referenced by:  numedglnl  28095
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