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Theorem usgredg2v 27001
 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 26999 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑦𝐴) → (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
54ralrimiva 3180 . . 3 (𝐺 ∈ USGraph → ∀𝑦𝐴 (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
65adantr 483 . 2 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ∀𝑦𝐴 (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
72usgrf1 26949 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→ran 𝐸)
87adantr 483 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐸:dom 𝐸1-1→ran 𝐸)
9 elrabi 3673 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} → 𝑦 ∈ dom 𝐸)
109, 3eleq2s 2929 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ dom 𝐸)
11 elrabi 3673 . . . . . . . . . 10 (𝑤 ∈ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} → 𝑤 ∈ dom 𝐸)
1211, 3eleq2s 2929 . . . . . . . . 9 (𝑤𝐴𝑤 ∈ dom 𝐸)
1310, 12anim12i 614 . . . . . . . 8 ((𝑦𝐴𝑤𝐴) → (𝑦 ∈ dom 𝐸𝑤 ∈ dom 𝐸))
14 f1fveq 7012 . . . . . . . 8 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ (𝑦 ∈ dom 𝐸𝑤 ∈ dom 𝐸)) → ((𝐸𝑦) = (𝐸𝑤) ↔ 𝑦 = 𝑤))
158, 13, 14syl2an 597 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → ((𝐸𝑦) = (𝐸𝑤) ↔ 𝑦 = 𝑤))
1615bicomd 225 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (𝑦 = 𝑤 ↔ (𝐸𝑦) = (𝐸𝑤)))
1716notbid 320 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ 𝑦 = 𝑤 ↔ ¬ (𝐸𝑦) = (𝐸𝑤)))
18 simpl 485 . . . . . . . . . 10 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐺 ∈ USGraph)
19 simpl 485 . . . . . . . . . 10 ((𝑦𝐴𝑤𝐴) → 𝑦𝐴)
2018, 19anim12i 614 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (𝐺 ∈ USGraph ∧ 𝑦𝐴))
21 preq1 4661 . . . . . . . . . . 11 (𝑢 = 𝑧 → {𝑢, 𝑁} = {𝑧, 𝑁})
2221eqeq2d 2830 . . . . . . . . . 10 (𝑢 = 𝑧 → ((𝐸𝑦) = {𝑢, 𝑁} ↔ (𝐸𝑦) = {𝑧, 𝑁}))
2322cbvriotavw 7116 . . . . . . . . 9 (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁})
241, 2, 3usgredg2vlem2 27000 . . . . . . . . 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 657 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (𝐺 ∈ USGraph ∧ 𝑤𝐴))
2721eqeq2d 2830 . . . . . . . . . 10 (𝑢 = 𝑧 → ((𝐸𝑤) = {𝑢, 𝑁} ↔ (𝐸𝑤) = {𝑧, 𝑁}))
2827cbvriotavw 7116 . . . . . . . . 9 (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})
291, 2, 3usgredg2vlem2 27000 . . . . . . . . 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 2835 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → ((𝐸𝑦) = (𝐸𝑤) ↔ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁}))
3231notbid 320 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ (𝐸𝑦) = (𝐸𝑤) ↔ ¬ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁}))
33 riotaex 7110 . . . . . . . . . . . 12 (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) ∈ V
3433a1i 11 . . . . . . . . . . 11 (𝑁𝑉 → (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) ∈ V)
35 id 22 . . . . . . . . . . 11 (𝑁𝑉𝑁𝑉)
36 riotaex 7110 . . . . . . . . . . . 12 (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∈ V
3736a1i 11 . . . . . . . . . . 11 (𝑁𝑉 → (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∈ V)
38 preq12bg 4776 . . . . . . . . . . 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 836 . . . . . . . . . 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 320 . . . . . . . . 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 484 . . . . . . . 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 980 . . . . . . . . . . 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 978 . . . . . . . . . . . . 13 (¬ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ↔ (¬ (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∨ ¬ 𝑁 = 𝑁))
4423, 28eqeq12i 2834 . . . . . . . . . . . . . . . . 17 ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ↔ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}))
4544notbii 322 . . . . . . . . . . . . . . . 16 (¬ (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ↔ ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}))
4645biimpi 218 . . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . . 15 𝑁 = 𝑁
4948pm2.24i 153 . . . . . . . . . . . . . 14 𝑁 = 𝑁 → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5047, 49jaoi 853 . . . . . . . . . . . . 13 ((¬ (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∨ ¬ 𝑁 = 𝑁) → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5143, 50sylbi 219 . . . . . . . . . . . 12 (¬ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5251adantr 483 . . . . . . . . . . 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 219 . . . . . . . . . 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 483 . . . . . . . 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 242 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (¬ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁} → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5756adantr 483 . . . . . 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 242 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ (𝐸𝑦) = (𝐸𝑤) → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5917, 58sylbid 242 . . . 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 3189 . 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 6672 . . . 4 (𝑦 = 𝑤 → ((𝐸𝑦) = {𝑧, 𝑁} ↔ (𝐸𝑤) = {𝑧, 𝑁}))
6463riotabidv 7108 . . 3 (𝑦 = 𝑤 → (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}))
6562, 64f1mpt 7011 . 2 (𝐹:𝐴1-1𝑉 ↔ (∀𝑦𝐴 (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉 ∧ ∀𝑦𝐴𝑤𝐴 ((𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤)))
666, 61, 65sylanbrc 585 1 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1𝑉)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1531   ∈ wcel 2108  ∀wral 3136  {crab 3140  Vcvv 3493  {cpr 4561   ↦ cmpt 5137  dom cdm 5548  ran crn 5549  –1-1→wf1 6345  ‘cfv 6348  ℩crio 7105  Vtxcvtx 26773  iEdgciedg 26774  USGraphcusgr 26926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885  df-hash 13683  df-edg 26825  df-umgr 26860  df-usgr 26928 This theorem is referenced by:  usgriedgleord  27002
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