MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgredg2v Structured version   Visualization version   GIF version

Theorem usgredg2v 28992
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 28990 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑦𝐴) → (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
54ralrimiva 3140 . . 3 (𝐺 ∈ USGraph → ∀𝑦𝐴 (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
65adantr 480 . 2 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → ∀𝑦𝐴 (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉)
72usgrf1 28940 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐸:dom 𝐸1-1→ran 𝐸)
87adantr 480 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐸:dom 𝐸1-1→ran 𝐸)
9 elrabi 3672 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} → 𝑦 ∈ dom 𝐸)
109, 3eleq2s 2845 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ dom 𝐸)
11 elrabi 3672 . . . . . . . . . 10 (𝑤 ∈ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)} → 𝑤 ∈ dom 𝐸)
1211, 3eleq2s 2845 . . . . . . . . 9 (𝑤𝐴𝑤 ∈ dom 𝐸)
1310, 12anim12i 612 . . . . . . . 8 ((𝑦𝐴𝑤𝐴) → (𝑦 ∈ dom 𝐸𝑤 ∈ dom 𝐸))
14 f1fveq 7257 . . . . . . . 8 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ (𝑦 ∈ dom 𝐸𝑤 ∈ dom 𝐸)) → ((𝐸𝑦) = (𝐸𝑤) ↔ 𝑦 = 𝑤))
158, 13, 14syl2an 595 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → ((𝐸𝑦) = (𝐸𝑤) ↔ 𝑦 = 𝑤))
1615bicomd 222 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (𝑦 = 𝑤 ↔ (𝐸𝑦) = (𝐸𝑤)))
1716notbid 318 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ 𝑦 = 𝑤 ↔ ¬ (𝐸𝑦) = (𝐸𝑤)))
18 simpl 482 . . . . . . . . . 10 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐺 ∈ USGraph)
19 simpl 482 . . . . . . . . . 10 ((𝑦𝐴𝑤𝐴) → 𝑦𝐴)
2018, 19anim12i 612 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (𝐺 ∈ USGraph ∧ 𝑦𝐴))
21 preq1 4732 . . . . . . . . . . 11 (𝑢 = 𝑧 → {𝑢, 𝑁} = {𝑧, 𝑁})
2221eqeq2d 2737 . . . . . . . . . 10 (𝑢 = 𝑧 → ((𝐸𝑦) = {𝑢, 𝑁} ↔ (𝐸𝑦) = {𝑧, 𝑁}))
2322cbvriotavw 7371 . . . . . . . . 9 (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁})
241, 2, 3usgredg2vlem2 28991 . . . . . . . . 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 656 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (𝐺 ∈ USGraph ∧ 𝑤𝐴))
2721eqeq2d 2737 . . . . . . . . . 10 (𝑢 = 𝑧 → ((𝐸𝑤) = {𝑢, 𝑁} ↔ (𝐸𝑤) = {𝑧, 𝑁}))
2827cbvriotavw 7371 . . . . . . . . 9 (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})
291, 2, 3usgredg2vlem2 28991 . . . . . . . . 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 2742 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → ((𝐸𝑦) = (𝐸𝑤) ↔ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁}))
3231notbid 318 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ (𝐸𝑦) = (𝐸𝑤) ↔ ¬ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁}))
33 riotaex 7365 . . . . . . . . . . . 12 (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) ∈ V
3433a1i 11 . . . . . . . . . . 11 (𝑁𝑉 → (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) ∈ V)
35 id 22 . . . . . . . . . . 11 (𝑁𝑉𝑁𝑉)
36 riotaex 7365 . . . . . . . . . . . 12 (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∈ V
3736a1i 11 . . . . . . . . . . 11 (𝑁𝑉 → (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∈ V)
38 preq12bg 4849 . . . . . . . . . . 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 318 . . . . . . . . 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 481 . . . . . . . 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 2744 . . . . . . . . . . . . . . . . 17 ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ↔ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}))
4544notbii 320 . . . . . . . . . . . . . . . 16 (¬ (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ↔ ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}))
4645biimpi 215 . . . . . . . . . . . . . . 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 2726 . . . . . . . . . . . . . . 15 𝑁 = 𝑁
4948pm2.24i 150 . . . . . . . . . . . . . 14 𝑁 = 𝑁 → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5047, 49jaoi 854 . . . . . . . . . . . . 13 ((¬ (𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∨ ¬ 𝑁 = 𝑁) → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5143, 50sylbi 216 . . . . . . . . . . . 12 (¬ ((𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}) = (𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}) ∧ 𝑁 = 𝑁) → (𝐺 ∈ USGraph → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5251adantr 480 . . . . . . . . . . 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 216 . . . . . . . . . 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 480 . . . . . . . 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 239 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → (¬ {(𝑢𝑉 (𝐸𝑦) = {𝑢, 𝑁}), 𝑁} = {(𝑢𝑉 (𝐸𝑤) = {𝑢, 𝑁}), 𝑁} → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5756adantr 480 . . . . . 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 239 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑤𝐴)) → (¬ (𝐸𝑦) = (𝐸𝑤) → ¬ (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁})))
5917, 58sylbid 239 . . . 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 3194 . 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 6894 . . . 4 (𝑦 = 𝑤 → ((𝐸𝑦) = {𝑧, 𝑁} ↔ (𝐸𝑤) = {𝑧, 𝑁}))
6463riotabidv 7363 . . 3 (𝑦 = 𝑤 → (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}))
6562, 64f1mpt 7256 . 2 (𝐹:𝐴1-1𝑉 ↔ (∀𝑦𝐴 (𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) ∈ 𝑉 ∧ ∀𝑦𝐴𝑤𝐴 ((𝑧𝑉 (𝐸𝑦) = {𝑧, 𝑁}) = (𝑧𝑉 (𝐸𝑤) = {𝑧, 𝑁}) → 𝑦 = 𝑤)))
666, 61, 65sylanbrc 582 1 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 844   = wceq 1533  wcel 2098  wral 3055  {crab 3426  Vcvv 3468  {cpr 4625  cmpt 5224  dom cdm 5669  ran crn 5670  1-1wf1 6534  cfv 6537  crio 7360  Vtxcvtx 28764  iEdgciedg 28765  USGraphcusgr 28917
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-2o 8468  df-oadd 8471  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-hash 14296  df-edg 28816  df-umgr 28851  df-usgr 28919
This theorem is referenced by:  usgriedgleord  28993
  Copyright terms: Public domain W3C validator