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Theorem edglnl 28976
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 5059 . . . 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 4163 . 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 4308 . . 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 481 . . . . . . . 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 4667 . . . . . . . 8 (𝑁𝑉𝑁 ∈ {𝑁})
98ad2antlr 725 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → 𝑁 ∈ {𝑁})
10 eleq2 2818 . . . . . . 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 28935 . . . . . . . . 9 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
14 edglnl.e . . . . . . . . . 10 𝐸 = (iEdg‘𝐺)
1514uhgrfun 28899 . . . . . . . . 9 (𝐺 ∈ UHGraph → Fun 𝐸)
1613, 15syl 17 . . . . . . . 8 (𝐺 ∈ UPGraph → Fun 𝐸)
1716adantr 479 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → Fun 𝐸)
1814iedgedg 28883 . . . . . . 7 ((Fun 𝐸𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
1917, 18sylan 578 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑖 ∈ dom 𝐸) → (𝐸𝑖) ∈ (Edg‘𝐺))
20 edglnl.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
21 eqid 2728 . . . . . . . . . 10 (Edg‘𝐺) = (Edg‘𝐺)
2220, 21upgredg 28970 . . . . . . . . 9 ((𝐺 ∈ UPGraph ∧ (𝐸𝑖) ∈ (Edg‘𝐺)) → ∃𝑛𝑉𝑚𝑉 (𝐸𝑖) = {𝑛, 𝑚})
2322ex 411 . . . . . . . 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 4645 . . . . . . . . . . . . . . . . . . . . . 22 {𝑛} = {𝑛, 𝑛}
2625eqcomi 2737 . . . . . . . . . . . . . . . . . . . . 21 {𝑛, 𝑛} = {𝑛}
27 elsni 4649 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ {𝑛} → 𝑁 = 𝑛)
28 sneq 4642 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 = 𝑛 → {𝑁} = {𝑛})
2928eqcomd 2734 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 = 𝑛 → {𝑛} = {𝑁})
3027, 29syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ {𝑛} → {𝑛} = {𝑁})
3126, 30eqtrid 2780 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ {𝑛} → {𝑛, 𝑛} = {𝑁})
3231, 26eleq2s 2847 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ {𝑛, 𝑛} → {𝑛, 𝑛} = {𝑁})
33 preq2 4743 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = 𝑛 → {𝑛, 𝑚} = {𝑛, 𝑛})
3433eleq2d 2815 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → (𝑁 ∈ {𝑛, 𝑚} ↔ 𝑁 ∈ {𝑛, 𝑛}))
3533eqeq1d 2730 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑛 → ({𝑛, 𝑚} = {𝑁} ↔ {𝑛, 𝑛} = {𝑁}))
3634, 35imbi12d 343 . . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 18 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑚𝑛)
4544necomd 2993 . . . . . . . . . . . . . . . . 17 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → 𝑛𝑚)
46 simpr2 1192 . . . . . . . . . . . . . . . . 17 ((𝑚𝑛 ∧ (𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚})) → (𝑛𝑉𝑚𝑉))
47 prproe 4910 . . . . . . . . . . . . . . . . 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 3188 . . . . . . . . . . . . . . . 16 (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ↔ (𝑁 ∈ {𝑛, 𝑚} ∧ ∃𝑣 ∈ (𝑉 ∖ {𝑁})𝑣 ∈ {𝑛, 𝑚}))
5043, 48, 49sylanbrc 581 . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . 13 (𝑚𝑛 → ((𝑁𝑉 ∧ (𝑛𝑉𝑚𝑉) ∧ 𝑁 ∈ {𝑛, 𝑚}) → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚}) ∨ {𝑛, 𝑚} = {𝑁})))
5342, 52pm2.61ine 3022 . . . . . . . . . . . 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 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 405 . . . . . . . . 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 2818 . . . . . . . . . 10 ((𝐸𝑖) = {𝑛, 𝑚} → (𝑁 ∈ (𝐸𝑖) ↔ 𝑁 ∈ {𝑛, 𝑚}))
58 eleq2 2818 . . . . . . . . . . . . 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 3176 . . . . . . . . . . 11 ((𝐸𝑖) = {𝑛, 𝑚} → (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ↔ ∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ {𝑛, 𝑚} ∧ 𝑣 ∈ {𝑛, 𝑚})))
61 eqeq1 2732 . . . . . . . . . . 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 343 . . . . . . . . 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 3209 . . . . . . 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 3437 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → {𝑖 ∈ dom 𝐸 ∣ (∃𝑣 ∈ (𝑉 ∖ {𝑁})(𝑁 ∈ (𝐸𝑖) ∧ 𝑣 ∈ (𝐸𝑖)) ∨ (𝐸𝑖) = {𝑁})} = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)})
704, 69eqtrid 2780 . 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 2768 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 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  wne 2937  wrex 3067  {crab 3430  cdif 3946  cun 3947  {csn 4632  {cpr 4634   ciun 5000  dom cdm 5682  Fun wfun 6547  cfv 6553  Vtxcvtx 28829  iEdgciedg 28830  Edgcedg 28880  UHGraphcuhgr 28889  UPGraphcupgr 28913
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-2o 8494  df-oadd 8497  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-dju 9932  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-n0 12511  df-xnn0 12583  df-z 12597  df-uz 12861  df-fz 13525  df-hash 14330  df-edg 28881  df-uhgr 28891  df-upgr 28915
This theorem is referenced by:  numedglnl  28977
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