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Theorem usgredg2v 27320
Description: In a simple graph, the mapping of edges having a fixed endpoint to the other vertex of the edge is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
Hypotheses
Ref Expression
usgredg2v.v 𝑉 = (Vtx‘𝐺)
usgredg2v.e 𝐸 = (iEdg‘𝐺)
usgredg2v.a 𝐴 = {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}
usgredg2v.f 𝐹 = (𝑦𝐴 ↦ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}))
Assertion
Ref Expression
usgredg2v ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1𝑉)
Distinct variable groups:   𝑥,𝐸,𝑧   𝑧,𝐺   𝑥,𝑁,𝑧   𝑧,𝑉   𝑦,𝐴   𝑦,𝐸,𝑥,𝑧   𝑦,𝐺   𝑦,𝑁   𝑦,𝑉
Allowed substitution hints:   𝐴(𝑥,𝑧)   𝐹(𝑥,𝑦,𝑧)   𝐺(𝑥)   𝑉(𝑥)

Proof of Theorem usgredg2v
Dummy variables 𝑤 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgredg2v.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 usgredg2v.e . . . . 5 𝐸 = (iEdg‘𝐺)
3 usgredg2v.a . . . . 5 𝐴 = {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}
41, 2, 3usgredg2vlem1 27318 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑦𝐴) → (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
54ralrimiva 3105 . . 3 (𝐺 ∈ USGraph → ∀𝑦𝐴 (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
65adantr 484 . 2 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ∀𝑦𝐴 (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
72usgrf1 27268 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→ran 𝐸)
87adantr 484 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐸:dom 𝐸1-1→ran 𝐸)
9 elrabi 3601 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} → 𝑦 ∈ dom 𝐸)
109, 3eleq2s 2856 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ dom 𝐸)
11 elrabi 3601 . . . . . . . . . 10 (𝑤 ∈ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} → 𝑤 ∈ dom 𝐸)
1211, 3eleq2s 2856 . . . . . . . . 9 (𝑤𝐴𝑤 ∈ dom 𝐸)
1310, 12anim12i 616 . . . . . . . 8 ((𝑦𝐴𝑤𝐴) → (𝑦 ∈ dom 𝐸𝑤 ∈ dom 𝐸))
14 f1fveq 7079 . . . . . . . 8 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ (𝑦 ∈ dom 𝐸𝑤 ∈ dom 𝐸)) → ((𝐸𝑦) = (𝐸𝑤) ↔ 𝑦 = 𝑤))
158, 13, 14syl2an 599 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → ((𝐸𝑦) = (𝐸𝑤) ↔ 𝑦 = 𝑤))
1615bicomd 226 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (𝑦 = 𝑤 ↔ (𝐸𝑦) = (𝐸𝑤)))
1716notbid 321 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ 𝑦 = 𝑤 ↔ ¬ (𝐸𝑦) = (𝐸𝑤)))
18 simpl 486 . . . . . . . . . 10 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐺 ∈ USGraph)
19 simpl 486 . . . . . . . . . 10 ((𝑦𝐴𝑤𝐴) → 𝑦𝐴)
2018, 19anim12i 616 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (𝐺 ∈ USGraph ∧ 𝑦𝐴))
21 preq1 4654 . . . . . . . . . . 11 (𝑢 = 𝑧 → {𝑢, 𝑁} = {𝑧, 𝑁})
2221eqeq2d 2748 . . . . . . . . . 10 (𝑢 = 𝑧 → ((𝐸𝑦) = {𝑢, 𝑁} ↔ (𝐸𝑦) = {𝑧, 𝑁}))
2322cbvriotavw 7185 . . . . . . . . 9 (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁})
241, 2, 3usgredg2vlem2 27319 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ 𝑦𝐴) → ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) → (𝐸𝑦) = {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁}))
2520, 23, 24mpisyl 21 . . . . . . . 8 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (𝐸𝑦) = {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁})
26 an3 659 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (𝐺 ∈ USGraph ∧ 𝑤𝐴))
2721eqeq2d 2748 . . . . . . . . . 10 (𝑢 = 𝑧 → ((𝐸𝑤) = {𝑢, 𝑁} ↔ (𝐸𝑤) = {𝑧, 𝑁}))
2827cbvriotavw 7185 . . . . . . . . 9 (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})
291, 2, 3usgredg2vlem2 27319 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ 𝑤𝐴) → ((𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}) → (𝐸𝑤) = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁}))
3026, 28, 29mpisyl 21 . . . . . . . 8 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (𝐸𝑤) = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁})
3125, 30eqeq12d 2753 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → ((𝐸𝑦) = (𝐸𝑤) ↔ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁}))
3231notbid 321 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ (𝐸𝑦) = (𝐸𝑤) ↔ ¬ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁}))
33 riotaex 7179 . . . . . . . . . . . 12 (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) ∈ V
3433a1i 11 . . . . . . . . . . 11 (𝑁𝑉 → (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) ∈ V)
35 id 22 . . . . . . . . . . 11 (𝑁𝑉𝑁𝑉)
36 riotaex 7179 . . . . . . . . . . . 12 (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∈ V
3736a1i 11 . . . . . . . . . . 11 (𝑁𝑉 → (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∈ V)
38 preq12bg 4769 . . . . . . . . . . 11 ((((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) ∈ V ∧ 𝑁𝑉) ∧ ((𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∈ V ∧ 𝑁𝑉)) → ({(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁} ↔ (((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = 𝑁𝑁 = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁})))))
3934, 35, 37, 35, 38syl22anc 839 . . . . . . . . . 10 (𝑁𝑉 → ({(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁} ↔ (((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = 𝑁𝑁 = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁})))))
4039notbid 321 . . . . . . . . 9 (𝑁𝑉 → (¬ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁} ↔ ¬ (((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = 𝑁𝑁 = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁})))))
4140adantl 485 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (¬ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁} ↔ ¬ (((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = 𝑁𝑁 = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁})))))
42 ioran 984 . . . . . . . . . . 11 (¬ (((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = 𝑁𝑁 = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}))) ↔ (¬ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∧ ¬ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = 𝑁𝑁 = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}))))
43 ianor 982 . . . . . . . . . . . . 13 (¬ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ↔ (¬ (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∨ ¬ 𝑁 = 𝑁))
4423, 28eqeq12i 2755 . . . . . . . . . . . . . . . . 17 ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ↔ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}))
4544notbii 323 . . . . . . . . . . . . . . . 16 (¬ (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ↔ ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}))
4645biimpi 219 . . . . . . . . . . . . . . 15 (¬ (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}))
4746a1d 25 . . . . . . . . . . . . . 14 (¬ (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
48 eqid 2737 . . . . . . . . . . . . . . 15 𝑁 = 𝑁
4948pm2.24i 153 . . . . . . . . . . . . . 14 𝑁 = 𝑁 → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5047, 49jaoi 857 . . . . . . . . . . . . 13 ((¬ (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∨ ¬ 𝑁 = 𝑁) → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5143, 50sylbi 220 . . . . . . . . . . . 12 (¬ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5251adantr 484 . . . . . . . . . . 11 ((¬ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∧ ¬ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = 𝑁𝑁 = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}))) → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5342, 52sylbi 220 . . . . . . . . . 10 (¬ (((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = 𝑁𝑁 = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}))) → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5453com12 32 . . . . . . . . 9 (𝐺 ∈ USGraph → (¬ (((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = 𝑁𝑁 = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}))) → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5554adantr 484 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (¬ (((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = 𝑁𝑁 = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}))) → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5641, 55sylbid 243 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (¬ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁} → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5756adantr 484 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁} → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5832, 57sylbid 243 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ (𝐸𝑦) = (𝐸𝑤) → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5917, 58sylbid 243 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ 𝑦 = 𝑤 → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
6059con4d 115 . . 3 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → ((𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤))
6160ralrimivva 3112 . 2 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ∀𝑦𝐴𝑤𝐴 ((𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤))
62 usgredg2v.f . . 3 𝐹 = (𝑦𝐴 ↦ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}))
63 fveqeq2 6731 . . . 4 (𝑦 = 𝑤 → ((𝐸𝑦) = {𝑧, 𝑁} ↔ (𝐸𝑤) = {𝑧, 𝑁}))
6463riotabidv 7177 . . 3 (𝑦 = 𝑤 → (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}))
6562, 64f1mpt 7078 . 2 (𝐹:𝐴1-1𝑉 ↔ (∀𝑦𝐴 (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉 ∧ ∀𝑦𝐴𝑤𝐴 ((𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤)))
666, 61, 65sylanbrc 586 1 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 847   = wceq 1543  wcel 2110  wral 3061  {crab 3065  Vcvv 3413  {cpr 4548  cmpt 5140  dom cdm 5556  ran crn 5557  1-1wf1 6382  cfv 6385  crio 7174  Vtxcvtx 27092  iEdgciedg 27093  USGraphcusgr 27245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5197  ax-nul 5204  ax-pow 5263  ax-pr 5327  ax-un 7528  ax-cnex 10790  ax-resscn 10791  ax-1cn 10792  ax-icn 10793  ax-addcl 10794  ax-addrcl 10795  ax-mulcl 10796  ax-mulrcl 10797  ax-mulcom 10798  ax-addass 10799  ax-mulass 10800  ax-distr 10801  ax-i2m1 10802  ax-1ne0 10803  ax-1rid 10804  ax-rnegex 10805  ax-rrecex 10806  ax-cnre 10807  ax-pre-lttri 10808  ax-pre-lttrn 10809  ax-pre-ltadd 10810  ax-pre-mulgt0 10811
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3415  df-sbc 3700  df-csb 3817  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-pss 3890  df-nul 4243  df-if 4445  df-pw 4520  df-sn 4547  df-pr 4549  df-tp 4551  df-op 4553  df-uni 4825  df-int 4865  df-iun 4911  df-br 5059  df-opab 5121  df-mpt 5141  df-tr 5167  df-id 5460  df-eprel 5465  df-po 5473  df-so 5474  df-fr 5514  df-we 5516  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-rn 5567  df-res 5568  df-ima 5569  df-pred 6165  df-ord 6221  df-on 6222  df-lim 6223  df-suc 6224  df-iota 6343  df-fun 6387  df-fn 6388  df-f 6389  df-f1 6390  df-fo 6391  df-f1o 6392  df-fv 6393  df-riota 7175  df-ov 7221  df-oprab 7222  df-mpo 7223  df-om 7650  df-1st 7766  df-2nd 7767  df-wrecs 8052  df-recs 8113  df-rdg 8151  df-1o 8207  df-2o 8208  df-oadd 8211  df-er 8396  df-en 8632  df-dom 8633  df-sdom 8634  df-fin 8635  df-dju 9522  df-card 9560  df-pnf 10874  df-mnf 10875  df-xr 10876  df-ltxr 10877  df-le 10878  df-sub 11069  df-neg 11070  df-nn 11836  df-2 11898  df-n0 12096  df-z 12182  df-uz 12444  df-fz 13101  df-hash 13902  df-edg 27144  df-umgr 27179  df-usgr 27247
This theorem is referenced by:  usgriedgleord  27321
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