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Theorem usgredg2v 26100
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 26098 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑦𝐴) → (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
54ralrimiva 2963 . . 3 (𝐺 ∈ USGraph → ∀𝑦𝐴 (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
65adantr 481 . 2 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ∀𝑦𝐴 (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
72usgrf1 26048 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→ran 𝐸)
87adantr 481 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐸:dom 𝐸1-1→ran 𝐸)
9 elrabi 3353 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} → 𝑦 ∈ dom 𝐸)
109, 3eleq2s 2717 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ dom 𝐸)
11 elrabi 3353 . . . . . . . . . 10 (𝑤 ∈ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} → 𝑤 ∈ dom 𝐸)
1211, 3eleq2s 2717 . . . . . . . . 9 (𝑤𝐴𝑤 ∈ dom 𝐸)
1310, 12anim12i 589 . . . . . . . 8 ((𝑦𝐴𝑤𝐴) → (𝑦 ∈ dom 𝐸𝑤 ∈ dom 𝐸))
14 f1fveq 6504 . . . . . . . 8 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ (𝑦 ∈ dom 𝐸𝑤 ∈ dom 𝐸)) → ((𝐸𝑦) = (𝐸𝑤) ↔ 𝑦 = 𝑤))
158, 13, 14syl2an 494 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → ((𝐸𝑦) = (𝐸𝑤) ↔ 𝑦 = 𝑤))
1615bicomd 213 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (𝑦 = 𝑤 ↔ (𝐸𝑦) = (𝐸𝑤)))
1716notbid 308 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ 𝑦 = 𝑤 ↔ ¬ (𝐸𝑦) = (𝐸𝑤)))
18 simpl 473 . . . . . . . . . 10 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐺 ∈ USGraph )
19 simpl 473 . . . . . . . . . 10 ((𝑦𝐴𝑤𝐴) → 𝑦𝐴)
2018, 19anim12i 589 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (𝐺 ∈ USGraph ∧ 𝑦𝐴))
21 preq1 4259 . . . . . . . . . . 11 (𝑢 = 𝑧 → {𝑢, 𝑁} = {𝑧, 𝑁})
2221eqeq2d 2630 . . . . . . . . . 10 (𝑢 = 𝑧 → ((𝐸𝑦) = {𝑢, 𝑁} ↔ (𝐸𝑦) = {𝑧, 𝑁}))
2322cbvriotav 6607 . . . . . . . . 9 (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁})
241, 2, 3usgredg2vlem2 26099 . . . . . . . . 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 simpr 477 . . . . . . . . . 10 ((𝑦𝐴𝑤𝐴) → 𝑤𝐴)
2718, 26anim12i 589 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (𝐺 ∈ USGraph ∧ 𝑤𝐴))
2821eqeq2d 2630 . . . . . . . . . 10 (𝑢 = 𝑧 → ((𝐸𝑤) = {𝑢, 𝑁} ↔ (𝐸𝑤) = {𝑧, 𝑁}))
2928cbvriotav 6607 . . . . . . . . 9 (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})
301, 2, 3usgredg2vlem2 26099 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ 𝑤𝐴) → ((𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}) → (𝐸𝑤) = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁}))
3127, 29, 30mpisyl 21 . . . . . . . 8 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (𝐸𝑤) = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁})
3225, 31eqeq12d 2635 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → ((𝐸𝑦) = (𝐸𝑤) ↔ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁}))
3332notbid 308 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ (𝐸𝑦) = (𝐸𝑤) ↔ ¬ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁}))
34 riotaex 6600 . . . . . . . . . . . 12 (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) ∈ V
3534a1i 11 . . . . . . . . . . 11 (𝑁𝑉 → (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) ∈ V)
36 id 22 . . . . . . . . . . 11 (𝑁𝑉𝑁𝑉)
37 riotaex 6600 . . . . . . . . . . . 12 (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∈ V
3837a1i 11 . . . . . . . . . . 11 (𝑁𝑉 → (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∈ V)
39 preq12bg 4377 . . . . . . . . . . 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})))))
4035, 36, 38, 36, 39syl22anc 1325 . . . . . . . . . 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})))))
4140notbid 308 . . . . . . . . 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})))))
4241adantl 482 . . . . . . . 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})))))
43 ioran 511 . . . . . . . . . . 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}))))
44 ianor 509 . . . . . . . . . . . . 13 (¬ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) ↔ (¬ (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∨ ¬ 𝑁 = 𝑁))
4523, 29eqeq12i 2634 . . . . . . . . . . . . . . . . 17 ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ↔ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}))
4645notbii 310 . . . . . . . . . . . . . . . 16 (¬ (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ↔ ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}))
4746biimpi 206 . . . . . . . . . . . . . . 15 (¬ (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}))
4847a1d 25 . . . . . . . . . . . . . 14 (¬ (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
49 eqid 2620 . . . . . . . . . . . . . . 15 𝑁 = 𝑁
5049pm2.24i 146 . . . . . . . . . . . . . 14 𝑁 = 𝑁 → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5148, 50jaoi 394 . . . . . . . . . . . . 13 ((¬ (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∨ ¬ 𝑁 = 𝑁) → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5244, 51sylbi 207 . . . . . . . . . . . 12 (¬ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5352adantr 481 . . . . . . . . . . 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})))
5443, 53sylbi 207 . . . . . . . . . 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})))
5554com12 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})))
5655adantr 481 . . . . . . . 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})))
5742, 56sylbid 230 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (¬ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁} → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5857adantr 481 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁} → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5933, 58sylbid 230 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ (𝐸𝑦) = (𝐸𝑤) → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
6017, 59sylbid 230 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ 𝑦 = 𝑤 → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
6160con4d 114 . . 3 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → ((𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤))
6261ralrimivva 2968 . 2 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ∀𝑦𝐴𝑤𝐴 ((𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤))
63 usgredg2v.f . . 3 𝐹 = (𝑦𝐴 ↦ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}))
64 fveq2 6178 . . . . 5 (𝑦 = 𝑤 → (𝐸𝑦) = (𝐸𝑤))
6564eqeq1d 2622 . . . 4 (𝑦 = 𝑤 → ((𝐸𝑦) = {𝑧, 𝑁} ↔ (𝐸𝑤) = {𝑧, 𝑁}))
6665riotabidv 6598 . . 3 (𝑦 = 𝑤 → (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}))
6763, 66f1mpt 6503 . 2 (𝐹:𝐴1-1𝑉 ↔ (∀𝑦𝐴 (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉 ∧ ∀𝑦𝐴𝑤𝐴 ((𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤)))
686, 62, 67sylanbrc 697 1 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1481  wcel 1988  wral 2909  {crab 2913  Vcvv 3195  {cpr 4170  cmpt 4720  dom cdm 5104  ran crn 5105  1-1wf1 5873  cfv 5876  crio 6595  Vtxcvtx 25855  iEdgciedg 25856   USGraph cusgr 26025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-hash 13101  df-edg 25921  df-umgr 25959  df-usgr 26027
This theorem is referenced by:  usgriedgleord  26101
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