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Theorem usgredg2v 28484
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 = (Vtxβ€˜πΊ)
usgredg2v.e 𝐸 = (iEdgβ€˜πΊ)
usgredg2v.a 𝐴 = {π‘₯ ∈ dom 𝐸 ∣ 𝑁 ∈ (πΈβ€˜π‘₯)}
usgredg2v.f 𝐹 = (𝑦 ∈ 𝐴 ↦ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}))
Assertion
Ref Expression
usgredg2v ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) β†’ 𝐹:𝐴–1-1→𝑉)
Distinct variable groups:   π‘₯,𝐸,𝑧   𝑧,𝐺   π‘₯,𝑁,𝑧   𝑧,𝑉   𝑦,𝐴   𝑦,𝐸,π‘₯,𝑧   𝑦,𝐺   𝑦,𝑁   𝑦,𝑉
Allowed substitution hints:   𝐴(π‘₯,𝑧)   𝐹(π‘₯,𝑦,𝑧)   𝐺(π‘₯)   𝑉(π‘₯)

Proof of Theorem usgredg2v
Dummy variables 𝑀 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgredg2v.v . . . . 5 𝑉 = (Vtxβ€˜πΊ)
2 usgredg2v.e . . . . 5 𝐸 = (iEdgβ€˜πΊ)
3 usgredg2v.a . . . . 5 𝐴 = {π‘₯ ∈ dom 𝐸 ∣ 𝑁 ∈ (πΈβ€˜π‘₯)}
41, 2, 3usgredg2vlem1 28482 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝐴) β†’ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) ∈ 𝑉)
54ralrimiva 3147 . . 3 (𝐺 ∈ USGraph β†’ βˆ€π‘¦ ∈ 𝐴 (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) ∈ 𝑉)
65adantr 482 . 2 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) β†’ βˆ€π‘¦ ∈ 𝐴 (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) ∈ 𝑉)
72usgrf1 28432 . . . . . . . . 9 (𝐺 ∈ USGraph β†’ 𝐸:dom 𝐸–1-1β†’ran 𝐸)
87adantr 482 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) β†’ 𝐸:dom 𝐸–1-1β†’ran 𝐸)
9 elrabi 3678 . . . . . . . . . 10 (𝑦 ∈ {π‘₯ ∈ dom 𝐸 ∣ 𝑁 ∈ (πΈβ€˜π‘₯)} β†’ 𝑦 ∈ dom 𝐸)
109, 3eleq2s 2852 . . . . . . . . 9 (𝑦 ∈ 𝐴 β†’ 𝑦 ∈ dom 𝐸)
11 elrabi 3678 . . . . . . . . . 10 (𝑀 ∈ {π‘₯ ∈ dom 𝐸 ∣ 𝑁 ∈ (πΈβ€˜π‘₯)} β†’ 𝑀 ∈ dom 𝐸)
1211, 3eleq2s 2852 . . . . . . . . 9 (𝑀 ∈ 𝐴 β†’ 𝑀 ∈ dom 𝐸)
1310, 12anim12i 614 . . . . . . . 8 ((𝑦 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴) β†’ (𝑦 ∈ dom 𝐸 ∧ 𝑀 ∈ dom 𝐸))
14 f1fveq 7261 . . . . . . . 8 ((𝐸:dom 𝐸–1-1β†’ran 𝐸 ∧ (𝑦 ∈ dom 𝐸 ∧ 𝑀 ∈ dom 𝐸)) β†’ ((πΈβ€˜π‘¦) = (πΈβ€˜π‘€) ↔ 𝑦 = 𝑀))
158, 13, 14syl2an 597 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ ((πΈβ€˜π‘¦) = (πΈβ€˜π‘€) ↔ 𝑦 = 𝑀))
1615bicomd 222 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ (𝑦 = 𝑀 ↔ (πΈβ€˜π‘¦) = (πΈβ€˜π‘€)))
1716notbid 318 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ (Β¬ 𝑦 = 𝑀 ↔ Β¬ (πΈβ€˜π‘¦) = (πΈβ€˜π‘€)))
18 simpl 484 . . . . . . . . . 10 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) β†’ 𝐺 ∈ USGraph)
19 simpl 484 . . . . . . . . . 10 ((𝑦 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴) β†’ 𝑦 ∈ 𝐴)
2018, 19anim12i 614 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ (𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝐴))
21 preq1 4738 . . . . . . . . . . 11 (𝑒 = 𝑧 β†’ {𝑒, 𝑁} = {𝑧, 𝑁})
2221eqeq2d 2744 . . . . . . . . . 10 (𝑒 = 𝑧 β†’ ((πΈβ€˜π‘¦) = {𝑒, 𝑁} ↔ (πΈβ€˜π‘¦) = {𝑧, 𝑁}))
2322cbvriotavw 7375 . . . . . . . . 9 (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁})
241, 2, 3usgredg2vlem2 28483 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ 𝑦 ∈ 𝐴) β†’ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) β†’ (πΈβ€˜π‘¦) = {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}), 𝑁}))
2520, 23, 24mpisyl 21 . . . . . . . 8 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ (πΈβ€˜π‘¦) = {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}), 𝑁})
26 an3 658 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ (𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝐴))
2721eqeq2d 2744 . . . . . . . . . 10 (𝑒 = 𝑧 β†’ ((πΈβ€˜π‘€) = {𝑒, 𝑁} ↔ (πΈβ€˜π‘€) = {𝑧, 𝑁}))
2827cbvriotavw 7375 . . . . . . . . 9 (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁})
291, 2, 3usgredg2vlem2 28483 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝐴) β†’ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁}) β†’ (πΈβ€˜π‘€) = {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}), 𝑁}))
3026, 28, 29mpisyl 21 . . . . . . . 8 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ (πΈβ€˜π‘€) = {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}), 𝑁})
3125, 30eqeq12d 2749 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ ((πΈβ€˜π‘¦) = (πΈβ€˜π‘€) ↔ {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}), 𝑁} = {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}), 𝑁}))
3231notbid 318 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ (Β¬ (πΈβ€˜π‘¦) = (πΈβ€˜π‘€) ↔ Β¬ {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}), 𝑁} = {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}), 𝑁}))
33 riotaex 7369 . . . . . . . . . . . 12 (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) ∈ V
3433a1i 11 . . . . . . . . . . 11 (𝑁 ∈ 𝑉 β†’ (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) ∈ V)
35 id 22 . . . . . . . . . . 11 (𝑁 ∈ 𝑉 β†’ 𝑁 ∈ 𝑉)
36 riotaex 7369 . . . . . . . . . . . 12 (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∈ V
3736a1i 11 . . . . . . . . . . 11 (𝑁 ∈ 𝑉 β†’ (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∈ V)
38 preq12bg 4855 . . . . . . . . . . 11 ((((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) ∈ V ∧ 𝑁 ∈ 𝑉) ∧ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∈ V ∧ 𝑁 ∈ 𝑉)) β†’ ({(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}), 𝑁} = {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}), 𝑁} ↔ (((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁})))))
3934, 35, 37, 35, 38syl22anc 838 . . . . . . . . . 10 (𝑁 ∈ 𝑉 β†’ ({(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}), 𝑁} = {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}), 𝑁} ↔ (((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁})))))
4039notbid 318 . . . . . . . . 9 (𝑁 ∈ 𝑉 β†’ (Β¬ {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}), 𝑁} = {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}), 𝑁} ↔ Β¬ (((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁})))))
4140adantl 483 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) β†’ (Β¬ {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}), 𝑁} = {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}), 𝑁} ↔ Β¬ (((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁})))))
42 ioran 983 . . . . . . . . . . 11 (Β¬ (((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}))) ↔ (Β¬ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∧ 𝑁 = 𝑁) ∧ Β¬ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}))))
43 ianor 981 . . . . . . . . . . . . 13 (Β¬ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∧ 𝑁 = 𝑁) ↔ (Β¬ (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∨ Β¬ 𝑁 = 𝑁))
4423, 28eqeq12i 2751 . . . . . . . . . . . . . . . . 17 ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ↔ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁}))
4544notbii 320 . . . . . . . . . . . . . . . 16 (Β¬ (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ↔ Β¬ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁}))
4645biimpi 215 . . . . . . . . . . . . . . 15 (Β¬ (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) β†’ Β¬ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁}))
4746a1d 25 . . . . . . . . . . . . . 14 (Β¬ (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) β†’ (𝐺 ∈ USGraph β†’ Β¬ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁})))
48 eqid 2733 . . . . . . . . . . . . . . 15 𝑁 = 𝑁
4948pm2.24i 150 . . . . . . . . . . . . . 14 (Β¬ 𝑁 = 𝑁 β†’ (𝐺 ∈ USGraph β†’ Β¬ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁})))
5047, 49jaoi 856 . . . . . . . . . . . . 13 ((Β¬ (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∨ Β¬ 𝑁 = 𝑁) β†’ (𝐺 ∈ USGraph β†’ Β¬ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁})))
5143, 50sylbi 216 . . . . . . . . . . . 12 (Β¬ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∧ 𝑁 = 𝑁) β†’ (𝐺 ∈ USGraph β†’ Β¬ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁})))
5251adantr 482 . . . . . . . . . . 11 ((Β¬ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∧ 𝑁 = 𝑁) ∧ Β¬ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}))) β†’ (𝐺 ∈ USGraph β†’ Β¬ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁})))
5342, 52sylbi 216 . . . . . . . . . 10 (Β¬ (((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}))) β†’ (𝐺 ∈ USGraph β†’ Β¬ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁})))
5453com12 32 . . . . . . . . 9 (𝐺 ∈ USGraph β†’ (Β¬ (((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}))) β†’ Β¬ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁})))
5554adantr 482 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) β†’ (Β¬ (((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}) ∧ 𝑁 = 𝑁) ∨ ((℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}) = 𝑁 ∧ 𝑁 = (℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}))) β†’ Β¬ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁})))
5641, 55sylbid 239 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) β†’ (Β¬ {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}), 𝑁} = {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}), 𝑁} β†’ Β¬ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁})))
5756adantr 482 . . . . . 6 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ (Β¬ {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑒, 𝑁}), 𝑁} = {(℩𝑒 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑒, 𝑁}), 𝑁} β†’ Β¬ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁})))
5832, 57sylbid 239 . . . . 5 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ (Β¬ (πΈβ€˜π‘¦) = (πΈβ€˜π‘€) β†’ Β¬ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁})))
5917, 58sylbid 239 . . . 4 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ (Β¬ 𝑦 = 𝑀 β†’ Β¬ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁})))
6059con4d 115 . . 3 (((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑦 ∈ 𝐴 ∧ 𝑀 ∈ 𝐴)) β†’ ((℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁}) β†’ 𝑦 = 𝑀))
6160ralrimivva 3201 . 2 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) β†’ βˆ€π‘¦ ∈ 𝐴 βˆ€π‘€ ∈ 𝐴 ((℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁}) β†’ 𝑦 = 𝑀))
62 usgredg2v.f . . 3 𝐹 = (𝑦 ∈ 𝐴 ↦ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}))
63 fveqeq2 6901 . . . 4 (𝑦 = 𝑀 β†’ ((πΈβ€˜π‘¦) = {𝑧, 𝑁} ↔ (πΈβ€˜π‘€) = {𝑧, 𝑁}))
6463riotabidv 7367 . . 3 (𝑦 = 𝑀 β†’ (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁}))
6562, 64f1mpt 7260 . 2 (𝐹:𝐴–1-1→𝑉 ↔ (βˆ€π‘¦ ∈ 𝐴 (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) ∈ 𝑉 ∧ βˆ€π‘¦ ∈ 𝐴 βˆ€π‘€ ∈ 𝐴 ((℩𝑧 ∈ 𝑉 (πΈβ€˜π‘¦) = {𝑧, 𝑁}) = (℩𝑧 ∈ 𝑉 (πΈβ€˜π‘€) = {𝑧, 𝑁}) β†’ 𝑦 = 𝑀)))
666, 61, 65sylanbrc 584 1 ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) β†’ 𝐹:𝐴–1-1→𝑉)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433  Vcvv 3475  {cpr 4631   ↦ cmpt 5232  dom cdm 5677  ran crn 5678  β€“1-1β†’wf1 6541  β€˜cfv 6544  β„©crio 7364  Vtxcvtx 28256  iEdgciedg 28257  USGraphcusgr 28409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-oadd 8470  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-hash 14291  df-edg 28308  df-umgr 28343  df-usgr 28411
This theorem is referenced by:  usgriedgleord  28485
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