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Theorem uhgr2edg 26299
Description: If a vertex is adjacent to two different vertices in a hypergraph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 11-Feb-2021.)
Hypotheses
Ref Expression
usgrf1oedg.i 𝐼 = (iEdg‘𝐺)
usgrf1oedg.e 𝐸 = (Edg‘𝐺)
uhgr2edg.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
uhgr2edg (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ (𝐴𝑉𝐵𝑉𝑁𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼(𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦)))
Distinct variable groups:   𝑥,𝐺   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐺   𝑥,𝐼,𝑦   𝑥,𝑁,𝑦   𝑥,𝑉,𝑦
Allowed substitution hints:   𝐸(𝑥,𝑦)

Proof of Theorem uhgr2edg
StepHypRef Expression
1 simp1l 1240 . . 3 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ (𝐴𝑉𝐵𝑉𝑁𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → 𝐺 ∈ UHGraph)
2 simp1r 1241 . . 3 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ (𝐴𝑉𝐵𝑉𝑁𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → 𝐴𝐵)
3 simp23 1251 . . . 4 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ (𝐴𝑉𝐵𝑉𝑁𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → 𝑁𝑉)
4 simp21 1249 . . . 4 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ (𝐴𝑉𝐵𝑉𝑁𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → 𝐴𝑉)
5 3simpc 1147 . . . . 5 ((𝐴𝑉𝐵𝑉𝑁𝑉) → (𝐵𝑉𝑁𝑉))
653ad2ant2 1129 . . . 4 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ (𝐴𝑉𝐵𝑉𝑁𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → (𝐵𝑉𝑁𝑉))
73, 4, 6jca31 558 . . 3 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ (𝐴𝑉𝐵𝑉𝑁𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉)))
81, 2, 7jca31 558 . 2 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ (𝐴𝑉𝐵𝑉𝑁𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))))
9 simp3 1133 . 2 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ (𝐴𝑉𝐵𝑉𝑁𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸))
10 usgrf1oedg.e . . . . . . . . 9 𝐸 = (Edg‘𝐺)
1110a1i 11 . . . . . . . 8 (𝐺 ∈ UHGraph → 𝐸 = (Edg‘𝐺))
12 edgval 26140 . . . . . . . . 9 (Edg‘𝐺) = ran (iEdg‘𝐺)
1312a1i 11 . . . . . . . 8 (𝐺 ∈ UHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
14 usgrf1oedg.i . . . . . . . . . . 11 𝐼 = (iEdg‘𝐺)
1514eqcomi 2769 . . . . . . . . . 10 (iEdg‘𝐺) = 𝐼
1615a1i 11 . . . . . . . . 9 (𝐺 ∈ UHGraph → (iEdg‘𝐺) = 𝐼)
1716rneqd 5508 . . . . . . . 8 (𝐺 ∈ UHGraph → ran (iEdg‘𝐺) = ran 𝐼)
1811, 13, 173eqtrd 2798 . . . . . . 7 (𝐺 ∈ UHGraph → 𝐸 = ran 𝐼)
1918eleq2d 2825 . . . . . 6 (𝐺 ∈ UHGraph → ({𝑁, 𝐴} ∈ 𝐸 ↔ {𝑁, 𝐴} ∈ ran 𝐼))
2018eleq2d 2825 . . . . . 6 (𝐺 ∈ UHGraph → ({𝐵, 𝑁} ∈ 𝐸 ↔ {𝐵, 𝑁} ∈ ran 𝐼))
2119, 20anbi12d 749 . . . . 5 (𝐺 ∈ UHGraph → (({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸) ↔ ({𝑁, 𝐴} ∈ ran 𝐼 ∧ {𝐵, 𝑁} ∈ ran 𝐼)))
2214uhgrfun 26160 . . . . . . 7 (𝐺 ∈ UHGraph → Fun 𝐼)
23 funfn 6079 . . . . . . 7 (Fun 𝐼𝐼 Fn dom 𝐼)
2422, 23sylib 208 . . . . . 6 (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
25 fvelrnb 6405 . . . . . . 7 (𝐼 Fn dom 𝐼 → ({𝑁, 𝐴} ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼(𝐼𝑥) = {𝑁, 𝐴}))
26 fvelrnb 6405 . . . . . . 7 (𝐼 Fn dom 𝐼 → ({𝐵, 𝑁} ∈ ran 𝐼 ↔ ∃𝑦 ∈ dom 𝐼(𝐼𝑦) = {𝐵, 𝑁}))
2725, 26anbi12d 749 . . . . . 6 (𝐼 Fn dom 𝐼 → (({𝑁, 𝐴} ∈ ran 𝐼 ∧ {𝐵, 𝑁} ∈ ran 𝐼) ↔ (∃𝑥 ∈ dom 𝐼(𝐼𝑥) = {𝑁, 𝐴} ∧ ∃𝑦 ∈ dom 𝐼(𝐼𝑦) = {𝐵, 𝑁})))
2824, 27syl 17 . . . . 5 (𝐺 ∈ UHGraph → (({𝑁, 𝐴} ∈ ran 𝐼 ∧ {𝐵, 𝑁} ∈ ran 𝐼) ↔ (∃𝑥 ∈ dom 𝐼(𝐼𝑥) = {𝑁, 𝐴} ∧ ∃𝑦 ∈ dom 𝐼(𝐼𝑦) = {𝐵, 𝑁})))
2921, 28bitrd 268 . . . 4 (𝐺 ∈ UHGraph → (({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸) ↔ (∃𝑥 ∈ dom 𝐼(𝐼𝑥) = {𝑁, 𝐴} ∧ ∃𝑦 ∈ dom 𝐼(𝐼𝑦) = {𝐵, 𝑁})))
3029ad2antrr 764 . . 3 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → (({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸) ↔ (∃𝑥 ∈ dom 𝐼(𝐼𝑥) = {𝑁, 𝐴} ∧ ∃𝑦 ∈ dom 𝐼(𝐼𝑦) = {𝐵, 𝑁})))
31 reeanv 3245 . . . 4 (∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼((𝐼𝑥) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁}) ↔ (∃𝑥 ∈ dom 𝐼(𝐼𝑥) = {𝑁, 𝐴} ∧ ∃𝑦 ∈ dom 𝐼(𝐼𝑦) = {𝐵, 𝑁}))
32 fveq2 6352 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝐼𝑥) = (𝐼𝑦))
3332eqeq1d 2762 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝐼𝑥) = {𝑁, 𝐴} ↔ (𝐼𝑦) = {𝑁, 𝐴}))
3433anbi1d 743 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝐼𝑥) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁}) ↔ ((𝐼𝑦) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁})))
35 eqtr2 2780 . . . . . . . . . . . . 13 (((𝐼𝑦) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁}) → {𝑁, 𝐴} = {𝐵, 𝑁})
36 prcom 4411 . . . . . . . . . . . . . . 15 {𝐵, 𝑁} = {𝑁, 𝐵}
3736eqeq2i 2772 . . . . . . . . . . . . . 14 ({𝑁, 𝐴} = {𝐵, 𝑁} ↔ {𝑁, 𝐴} = {𝑁, 𝐵})
38 preq12bg 4530 . . . . . . . . . . . . . . . . . 18 (((𝑁𝑉𝐴𝑉) ∧ (𝑁𝑉𝐵𝑉)) → ({𝑁, 𝐴} = {𝑁, 𝐵} ↔ ((𝑁 = 𝑁𝐴 = 𝐵) ∨ (𝑁 = 𝐵𝐴 = 𝑁))))
3938ancom2s 879 . . . . . . . . . . . . . . . . 17 (((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉)) → ({𝑁, 𝐴} = {𝑁, 𝐵} ↔ ((𝑁 = 𝑁𝐴 = 𝐵) ∨ (𝑁 = 𝐵𝐴 = 𝑁))))
40 eqneqall 2943 . . . . . . . . . . . . . . . . . . . 20 (𝐴 = 𝐵 → (𝐴𝐵𝑥𝑦))
4140adantl 473 . . . . . . . . . . . . . . . . . . 19 ((𝑁 = 𝑁𝐴 = 𝐵) → (𝐴𝐵𝑥𝑦))
42 eqtr 2779 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 = 𝑁𝑁 = 𝐵) → 𝐴 = 𝐵)
4342ancoms 468 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 = 𝐵𝐴 = 𝑁) → 𝐴 = 𝐵)
4443, 40syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑁 = 𝐵𝐴 = 𝑁) → (𝐴𝐵𝑥𝑦))
4541, 44jaoi 393 . . . . . . . . . . . . . . . . . 18 (((𝑁 = 𝑁𝐴 = 𝐵) ∨ (𝑁 = 𝐵𝐴 = 𝑁)) → (𝐴𝐵𝑥𝑦))
4645adantld 484 . . . . . . . . . . . . . . . . 17 (((𝑁 = 𝑁𝐴 = 𝐵) ∨ (𝑁 = 𝐵𝐴 = 𝑁)) → ((𝐺 ∈ UHGraph ∧ 𝐴𝐵) → 𝑥𝑦))
4739, 46syl6bi 243 . . . . . . . . . . . . . . . 16 (((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉)) → ({𝑁, 𝐴} = {𝑁, 𝐵} → ((𝐺 ∈ UHGraph ∧ 𝐴𝐵) → 𝑥𝑦)))
4847com3l 89 . . . . . . . . . . . . . . 15 ({𝑁, 𝐴} = {𝑁, 𝐵} → ((𝐺 ∈ UHGraph ∧ 𝐴𝐵) → (((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉)) → 𝑥𝑦)))
4948impd 446 . . . . . . . . . . . . . 14 ({𝑁, 𝐴} = {𝑁, 𝐵} → (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → 𝑥𝑦))
5037, 49sylbi 207 . . . . . . . . . . . . 13 ({𝑁, 𝐴} = {𝐵, 𝑁} → (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → 𝑥𝑦))
5135, 50syl 17 . . . . . . . . . . . 12 (((𝐼𝑦) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁}) → (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → 𝑥𝑦))
5234, 51syl6bi 243 . . . . . . . . . . 11 (𝑥 = 𝑦 → (((𝐼𝑥) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁}) → (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → 𝑥𝑦)))
5352com23 86 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → (((𝐼𝑥) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁}) → 𝑥𝑦)))
5453impd 446 . . . . . . . . 9 (𝑥 = 𝑦 → ((((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) ∧ ((𝐼𝑥) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁})) → 𝑥𝑦))
55 ax-1 6 . . . . . . . . 9 (𝑥𝑦 → ((((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) ∧ ((𝐼𝑥) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁})) → 𝑥𝑦))
5654, 55pm2.61ine 3015 . . . . . . . 8 ((((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) ∧ ((𝐼𝑥) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁})) → 𝑥𝑦)
57 prid1g 4439 . . . . . . . . . . . . 13 (𝑁𝑉𝑁 ∈ {𝑁, 𝐴})
5857ad2antrr 764 . . . . . . . . . . . 12 (((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉)) → 𝑁 ∈ {𝑁, 𝐴})
5958adantl 473 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → 𝑁 ∈ {𝑁, 𝐴})
60 eleq2 2828 . . . . . . . . . . 11 ((𝐼𝑥) = {𝑁, 𝐴} → (𝑁 ∈ (𝐼𝑥) ↔ 𝑁 ∈ {𝑁, 𝐴}))
6159, 60syl5ibr 236 . . . . . . . . . 10 ((𝐼𝑥) = {𝑁, 𝐴} → (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → 𝑁 ∈ (𝐼𝑥)))
6261adantr 472 . . . . . . . . 9 (((𝐼𝑥) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁}) → (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → 𝑁 ∈ (𝐼𝑥)))
6362impcom 445 . . . . . . . 8 ((((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) ∧ ((𝐼𝑥) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁})) → 𝑁 ∈ (𝐼𝑥))
64 prid2g 4440 . . . . . . . . . . . . 13 (𝑁𝑉𝑁 ∈ {𝐵, 𝑁})
6564ad2antrr 764 . . . . . . . . . . . 12 (((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉)) → 𝑁 ∈ {𝐵, 𝑁})
6665adantl 473 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → 𝑁 ∈ {𝐵, 𝑁})
67 eleq2 2828 . . . . . . . . . . 11 ((𝐼𝑦) = {𝐵, 𝑁} → (𝑁 ∈ (𝐼𝑦) ↔ 𝑁 ∈ {𝐵, 𝑁}))
6866, 67syl5ibr 236 . . . . . . . . . 10 ((𝐼𝑦) = {𝐵, 𝑁} → (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → 𝑁 ∈ (𝐼𝑦)))
6968adantl 473 . . . . . . . . 9 (((𝐼𝑥) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁}) → (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → 𝑁 ∈ (𝐼𝑦)))
7069impcom 445 . . . . . . . 8 ((((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) ∧ ((𝐼𝑥) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁})) → 𝑁 ∈ (𝐼𝑦))
7156, 63, 703jca 1123 . . . . . . 7 ((((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) ∧ ((𝐼𝑥) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁})) → (𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦)))
7271ex 449 . . . . . 6 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → (((𝐼𝑥) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁}) → (𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦))))
7372reximdv 3154 . . . . 5 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → (∃𝑦 ∈ dom 𝐼((𝐼𝑥) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁}) → ∃𝑦 ∈ dom 𝐼(𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦))))
7473reximdv 3154 . . . 4 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → (∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼((𝐼𝑥) = {𝑁, 𝐴} ∧ (𝐼𝑦) = {𝐵, 𝑁}) → ∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼(𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦))))
7531, 74syl5bir 233 . . 3 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → ((∃𝑥 ∈ dom 𝐼(𝐼𝑥) = {𝑁, 𝐴} ∧ ∃𝑦 ∈ dom 𝐼(𝐼𝑦) = {𝐵, 𝑁}) → ∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼(𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦))))
7630, 75sylbid 230 . 2 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ ((𝑁𝑉𝐴𝑉) ∧ (𝐵𝑉𝑁𝑉))) → (({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸) → ∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼(𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦))))
778, 9, 76sylc 65 1 (((𝐺 ∈ UHGraph ∧ 𝐴𝐵) ∧ (𝐴𝑉𝐵𝑉𝑁𝑉) ∧ ({𝑁, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝑁} ∈ 𝐸)) → ∃𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼(𝑥𝑦𝑁 ∈ (𝐼𝑥) ∧ 𝑁 ∈ (𝐼𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wrex 3051  {cpr 4323  dom cdm 5266  ran crn 5267  Fun wfun 6043   Fn wfn 6044  cfv 6049  Vtxcvtx 26073  iEdgciedg 26074  Edgcedg 26138  UHGraphcuhgr 26150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-edg 26139  df-uhgr 26152
This theorem is referenced by:  umgr2edg  26300
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