Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nbgraf1olem5 Structured version   Visualization version   GIF version

Theorem nbgraf1olem5 25974
 Description: Lemma 5 for nbgraf1o 25976. The mapping of neighbors to edge indices is a one-to-one onto function. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
Hypotheses
Ref Expression
nbgraf1o.n 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)
nbgraf1o.i 𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}
nbgraf1o.f 𝐹 = (𝑛𝑁 ↦ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))
Assertion
Ref Expression
nbgraf1olem5 ((𝑉 USGrph 𝐸𝑈𝑉) → 𝐹:𝑁1-1-onto𝐼)
Distinct variable groups:   𝑖,𝐸,𝑛   𝑈,𝑖,𝑛   𝑖,𝑉,𝑛   𝑖,𝐼,𝑛   𝑛,𝑁
Allowed substitution hints:   𝐹(𝑖,𝑛)   𝑁(𝑖)

Proof of Theorem nbgraf1olem5
Dummy variables 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nbgraf1o.n . . . 4 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)
2 nbgraf1o.i . . . 4 𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}
3 nbgraf1o.f . . . 4 𝐹 = (𝑛𝑁 ↦ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))
41, 2, 3nbgraf1olem2 25971 . . 3 ((𝑉 USGrph 𝐸𝑈𝑉) → 𝐹:𝑁𝐼)
5 ffn 5958 . . 3 (𝐹:𝑁𝐼𝐹 Fn 𝑁)
64, 5syl 17 . 2 ((𝑉 USGrph 𝐸𝑈𝑉) → 𝐹 Fn 𝑁)
71, 2, 3nbgraf1olem3 25972 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸𝑈𝑉𝑛𝑁) → (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}) = (𝐸‘{𝑈, 𝑛}))
873expa 1257 . . . . . . . . . 10 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑛𝑁) → (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}) = (𝐸‘{𝑈, 𝑛}))
98eqeq2d 2620 . . . . . . . . 9 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑛𝑁) → (𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}) ↔ 𝑗 = (𝐸‘{𝑈, 𝑛})))
10 usgraf1o 25887 . . . . . . . . . . . . . 14 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1-onto→ran 𝐸)
1110adantr 480 . . . . . . . . . . . . 13 ((𝑉 USGrph 𝐸𝑈𝑉) → 𝐸:dom 𝐸1-1-onto→ran 𝐸)
1211adantr 480 . . . . . . . . . . . 12 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑛𝑁) → 𝐸:dom 𝐸1-1-onto→ran 𝐸)
13 nbgraeledg 25959 . . . . . . . . . . . . . . . . . 18 (𝑉 USGrph 𝐸 → (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑛, 𝑈} ∈ ran 𝐸))
14 prcom 4211 . . . . . . . . . . . . . . . . . . 19 {𝑛, 𝑈} = {𝑈, 𝑛}
1514eleq1i 2679 . . . . . . . . . . . . . . . . . 18 ({𝑛, 𝑈} ∈ ran 𝐸 ↔ {𝑈, 𝑛} ∈ ran 𝐸)
1613, 15syl6bb 275 . . . . . . . . . . . . . . . . 17 (𝑉 USGrph 𝐸 → (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑈, 𝑛} ∈ ran 𝐸))
1716biimpcd 238 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) → (𝑉 USGrph 𝐸 → {𝑈, 𝑛} ∈ ran 𝐸))
1817a1d 25 . . . . . . . . . . . . . . 15 (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) → (𝑈𝑉 → (𝑉 USGrph 𝐸 → {𝑈, 𝑛} ∈ ran 𝐸)))
1918, 1eleq2s 2706 . . . . . . . . . . . . . 14 (𝑛𝑁 → (𝑈𝑉 → (𝑉 USGrph 𝐸 → {𝑈, 𝑛} ∈ ran 𝐸)))
2019com13 86 . . . . . . . . . . . . 13 (𝑉 USGrph 𝐸 → (𝑈𝑉 → (𝑛𝑁 → {𝑈, 𝑛} ∈ ran 𝐸)))
2120imp31 447 . . . . . . . . . . . 12 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑛𝑁) → {𝑈, 𝑛} ∈ ran 𝐸)
22 f1ocnvdm 6440 . . . . . . . . . . . 12 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ {𝑈, 𝑛} ∈ ran 𝐸) → (𝐸‘{𝑈, 𝑛}) ∈ dom 𝐸)
2312, 21, 22syl2anc 691 . . . . . . . . . . 11 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑛𝑁) → (𝐸‘{𝑈, 𝑛}) ∈ dom 𝐸)
24 prid1g 4239 . . . . . . . . . . . . . 14 (𝑈𝑉𝑈 ∈ {𝑈, 𝑛})
2524adantl 481 . . . . . . . . . . . . 13 ((𝑉 USGrph 𝐸𝑈𝑉) → 𝑈 ∈ {𝑈, 𝑛})
2625adantr 480 . . . . . . . . . . . 12 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑛𝑁) → 𝑈 ∈ {𝑈, 𝑛})
271eleq2i 2680 . . . . . . . . . . . . . 14 (𝑛𝑁𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈))
28 nbgracnvfv 25969 . . . . . . . . . . . . . 14 ((𝑉 USGrph 𝐸𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈)) → (𝐸‘(𝐸‘{𝑈, 𝑛})) = {𝑈, 𝑛})
2927, 28sylan2b 491 . . . . . . . . . . . . 13 ((𝑉 USGrph 𝐸𝑛𝑁) → (𝐸‘(𝐸‘{𝑈, 𝑛})) = {𝑈, 𝑛})
3029adantlr 747 . . . . . . . . . . . 12 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑛𝑁) → (𝐸‘(𝐸‘{𝑈, 𝑛})) = {𝑈, 𝑛})
3126, 30eleqtrrd 2691 . . . . . . . . . . 11 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑛𝑁) → 𝑈 ∈ (𝐸‘(𝐸‘{𝑈, 𝑛})))
3223, 31jca 553 . . . . . . . . . 10 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑛𝑁) → ((𝐸‘{𝑈, 𝑛}) ∈ dom 𝐸𝑈 ∈ (𝐸‘(𝐸‘{𝑈, 𝑛}))))
33 eleq1 2676 . . . . . . . . . . 11 (𝑗 = (𝐸‘{𝑈, 𝑛}) → (𝑗 ∈ dom 𝐸 ↔ (𝐸‘{𝑈, 𝑛}) ∈ dom 𝐸))
34 fveq2 6103 . . . . . . . . . . . 12 (𝑗 = (𝐸‘{𝑈, 𝑛}) → (𝐸𝑗) = (𝐸‘(𝐸‘{𝑈, 𝑛})))
3534eleq2d 2673 . . . . . . . . . . 11 (𝑗 = (𝐸‘{𝑈, 𝑛}) → (𝑈 ∈ (𝐸𝑗) ↔ 𝑈 ∈ (𝐸‘(𝐸‘{𝑈, 𝑛}))))
3633, 35anbi12d 743 . . . . . . . . . 10 (𝑗 = (𝐸‘{𝑈, 𝑛}) → ((𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗)) ↔ ((𝐸‘{𝑈, 𝑛}) ∈ dom 𝐸𝑈 ∈ (𝐸‘(𝐸‘{𝑈, 𝑛})))))
3732, 36syl5ibrcom 236 . . . . . . . . 9 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑛𝑁) → (𝑗 = (𝐸‘{𝑈, 𝑛}) → (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))))
389, 37sylbid 229 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑛𝑁) → (𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}) → (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))))
3938rexlimdva 3013 . . . . . . 7 ((𝑉 USGrph 𝐸𝑈𝑉) → (∃𝑛𝑁 𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}) → (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))))
40 simpl 472 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸𝑈𝑉) → 𝑉 USGrph 𝐸)
4140adantr 480 . . . . . . . . . 10 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) → 𝑉 USGrph 𝐸)
42 simpl 472 . . . . . . . . . . 11 ((𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗)) → 𝑗 ∈ dom 𝐸)
4342adantl 481 . . . . . . . . . 10 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) → 𝑗 ∈ dom 𝐸)
44 simprr 792 . . . . . . . . . 10 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) → 𝑈 ∈ (𝐸𝑗))
45 usgraedg4 25916 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗)) → ∃𝑛𝑉 (𝐸𝑗) = {𝑈, 𝑛})
4641, 43, 44, 45syl3anc 1318 . . . . . . . . 9 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) → ∃𝑛𝑉 (𝐸𝑗) = {𝑈, 𝑛})
47 usgrafun 25878 . . . . . . . . . . . . . . . . . . . . 21 (𝑉 USGrph 𝐸 → Fun 𝐸)
48 fvelrn 6260 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun 𝐸𝑗 ∈ dom 𝐸) → (𝐸𝑗) ∈ ran 𝐸)
4948ex 449 . . . . . . . . . . . . . . . . . . . . 21 (Fun 𝐸 → (𝑗 ∈ dom 𝐸 → (𝐸𝑗) ∈ ran 𝐸))
5047, 49syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑉 USGrph 𝐸 → (𝑗 ∈ dom 𝐸 → (𝐸𝑗) ∈ ran 𝐸))
5150adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸𝑈𝑉) → (𝑗 ∈ dom 𝐸 → (𝐸𝑗) ∈ ran 𝐸))
5251com12 32 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ dom 𝐸 → ((𝑉 USGrph 𝐸𝑈𝑉) → (𝐸𝑗) ∈ ran 𝐸))
5352adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗)) → ((𝑉 USGrph 𝐸𝑈𝑉) → (𝐸𝑗) ∈ ran 𝐸))
5453impcom 445 . . . . . . . . . . . . . . . 16 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) → (𝐸𝑗) ∈ ran 𝐸)
5554adantr 480 . . . . . . . . . . . . . . 15 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) ∧ (𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛})) → (𝐸𝑗) ∈ ran 𝐸)
56 id 22 . . . . . . . . . . . . . . . . . . 19 ({𝑈, 𝑛} = (𝐸𝑗) → {𝑈, 𝑛} = (𝐸𝑗))
5756eqcoms 2618 . . . . . . . . . . . . . . . . . 18 ((𝐸𝑗) = {𝑈, 𝑛} → {𝑈, 𝑛} = (𝐸𝑗))
5814, 57syl5eq 2656 . . . . . . . . . . . . . . . . 17 ((𝐸𝑗) = {𝑈, 𝑛} → {𝑛, 𝑈} = (𝐸𝑗))
5958eleq1d 2672 . . . . . . . . . . . . . . . 16 ((𝐸𝑗) = {𝑈, 𝑛} → ({𝑛, 𝑈} ∈ ran 𝐸 ↔ (𝐸𝑗) ∈ ran 𝐸))
6059ad2antll 761 . . . . . . . . . . . . . . 15 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) ∧ (𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛})) → ({𝑛, 𝑈} ∈ ran 𝐸 ↔ (𝐸𝑗) ∈ ran 𝐸))
6155, 60mpbird 246 . . . . . . . . . . . . . 14 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) ∧ (𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛})) → {𝑛, 𝑈} ∈ ran 𝐸)
6213ad3antrrr 762 . . . . . . . . . . . . . 14 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) ∧ (𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛})) → (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑛, 𝑈} ∈ ran 𝐸))
6361, 62mpbird 246 . . . . . . . . . . . . 13 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) ∧ (𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛})) → 𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈))
6463, 27sylibr 223 . . . . . . . . . . . 12 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) ∧ (𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛})) → 𝑛𝑁)
65 f1ocnvfv1 6432 . . . . . . . . . . . . . . . 16 ((𝐸:dom 𝐸1-1-onto→ran 𝐸𝑗 ∈ dom 𝐸) → (𝐸‘(𝐸𝑗)) = 𝑗)
6611, 42, 65syl2an 493 . . . . . . . . . . . . . . 15 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) → (𝐸‘(𝐸𝑗)) = 𝑗)
6766adantr 480 . . . . . . . . . . . . . 14 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) ∧ (𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛})) → (𝐸‘(𝐸𝑗)) = 𝑗)
68 fveq2 6103 . . . . . . . . . . . . . . . . 17 ({𝑈, 𝑛} = (𝐸𝑗) → (𝐸‘{𝑈, 𝑛}) = (𝐸‘(𝐸𝑗)))
6968eqcoms 2618 . . . . . . . . . . . . . . . 16 ((𝐸𝑗) = {𝑈, 𝑛} → (𝐸‘{𝑈, 𝑛}) = (𝐸‘(𝐸𝑗)))
7069eqeq1d 2612 . . . . . . . . . . . . . . 15 ((𝐸𝑗) = {𝑈, 𝑛} → ((𝐸‘{𝑈, 𝑛}) = 𝑗 ↔ (𝐸‘(𝐸𝑗)) = 𝑗))
7170ad2antll 761 . . . . . . . . . . . . . 14 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) ∧ (𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛})) → ((𝐸‘{𝑈, 𝑛}) = 𝑗 ↔ (𝐸‘(𝐸𝑗)) = 𝑗))
7267, 71mpbird 246 . . . . . . . . . . . . 13 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) ∧ (𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛})) → (𝐸‘{𝑈, 𝑛}) = 𝑗)
73 eqcom 2617 . . . . . . . . . . . . . 14 (𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}) ↔ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}) = 𝑗)
74 simplll 794 . . . . . . . . . . . . . . 15 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) ∧ (𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛})) → 𝑉 USGrph 𝐸)
75 simpllr 795 . . . . . . . . . . . . . . 15 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) ∧ (𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛})) → 𝑈𝑉)
767eqeq1d 2612 . . . . . . . . . . . . . . 15 ((𝑉 USGrph 𝐸𝑈𝑉𝑛𝑁) → ((𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}) = 𝑗 ↔ (𝐸‘{𝑈, 𝑛}) = 𝑗))
7774, 75, 64, 76syl3anc 1318 . . . . . . . . . . . . . 14 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) ∧ (𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛})) → ((𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}) = 𝑗 ↔ (𝐸‘{𝑈, 𝑛}) = 𝑗))
7873, 77syl5bb 271 . . . . . . . . . . . . 13 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) ∧ (𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛})) → (𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}) ↔ (𝐸‘{𝑈, 𝑛}) = 𝑗))
7972, 78mpbird 246 . . . . . . . . . . . 12 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) ∧ (𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛})) → 𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))
8064, 79jca 553 . . . . . . . . . . 11 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) ∧ (𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛})) → (𝑛𝑁𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛})))
8180ex 449 . . . . . . . . . 10 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) → ((𝑛𝑉 ∧ (𝐸𝑗) = {𝑈, 𝑛}) → (𝑛𝑁𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))))
8281reximdv2 2997 . . . . . . . . 9 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) → (∃𝑛𝑉 (𝐸𝑗) = {𝑈, 𝑛} → ∃𝑛𝑁 𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛})))
8346, 82mpd 15 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))) → ∃𝑛𝑁 𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))
8483ex 449 . . . . . . 7 ((𝑉 USGrph 𝐸𝑈𝑉) → ((𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗)) → ∃𝑛𝑁 𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛})))
8539, 84impbid 201 . . . . . 6 ((𝑉 USGrph 𝐸𝑈𝑉) → (∃𝑛𝑁 𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}) ↔ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))))
8685abbidv 2728 . . . . 5 ((𝑉 USGrph 𝐸𝑈𝑉) → {𝑗 ∣ ∃𝑛𝑁 𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛})} = {𝑗 ∣ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))})
87 eleq1 2676 . . . . . . 7 (𝑗 = 𝑖 → (𝑗 ∈ dom 𝐸𝑖 ∈ dom 𝐸))
88 fveq2 6103 . . . . . . . 8 (𝑗 = 𝑖 → (𝐸𝑗) = (𝐸𝑖))
8988eleq2d 2673 . . . . . . 7 (𝑗 = 𝑖 → (𝑈 ∈ (𝐸𝑗) ↔ 𝑈 ∈ (𝐸𝑖)))
9087, 89anbi12d 743 . . . . . 6 (𝑗 = 𝑖 → ((𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗)) ↔ (𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖))))
9190cbvabv 2734 . . . . 5 {𝑗 ∣ (𝑗 ∈ dom 𝐸𝑈 ∈ (𝐸𝑗))} = {𝑖 ∣ (𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖))}
9286, 91syl6eq 2660 . . . 4 ((𝑉 USGrph 𝐸𝑈𝑉) → {𝑗 ∣ ∃𝑛𝑁 𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛})} = {𝑖 ∣ (𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖))})
93 df-rab 2905 . . . 4 {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)} = {𝑖 ∣ (𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖))}
9492, 93syl6eqr 2662 . . 3 ((𝑉 USGrph 𝐸𝑈𝑉) → {𝑗 ∣ ∃𝑛𝑁 𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛})} = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)})
953rnmpt 5292 . . 3 ran 𝐹 = {𝑗 ∣ ∃𝑛𝑁 𝑗 = (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛})}
9694, 95, 23eqtr4g 2669 . 2 ((𝑉 USGrph 𝐸𝑈𝑉) → ran 𝐹 = 𝐼)
971, 2, 3nbgraf1olem4 25973 . . . . . . . 8 ((𝑉 USGrph 𝐸𝑈𝑉𝑥𝑁) → (𝐹𝑥) = (𝐸‘{𝑈, 𝑥}))
98973expa 1257 . . . . . . 7 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑥𝑁) → (𝐹𝑥) = (𝐸‘{𝑈, 𝑥}))
9998adantr 480 . . . . . 6 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → (𝐹𝑥) = (𝐸‘{𝑈, 𝑥}))
100 simplll 794 . . . . . . 7 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → 𝑉 USGrph 𝐸)
101 simpllr 795 . . . . . . 7 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → 𝑈𝑉)
102 simpr 476 . . . . . . 7 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → 𝑦𝑁)
1031, 2, 3nbgraf1olem4 25973 . . . . . . 7 ((𝑉 USGrph 𝐸𝑈𝑉𝑦𝑁) → (𝐹𝑦) = (𝐸‘{𝑈, 𝑦}))
104100, 101, 102, 103syl3anc 1318 . . . . . 6 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → (𝐹𝑦) = (𝐸‘{𝑈, 𝑦}))
10599, 104eqeq12d 2625 . . . . 5 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐸‘{𝑈, 𝑥}) = (𝐸‘{𝑈, 𝑦})))
106 usgraf1 25889 . . . . . . 7 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→ran 𝐸)
107106ad3antrrr 762 . . . . . 6 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → 𝐸:dom 𝐸1-1→ran 𝐸)
1081eleq2i 2680 . . . . . . . . . 10 (𝑥𝑁𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈))
109 nbgraeledg 25959 . . . . . . . . . . . 12 (𝑉 USGrph 𝐸 → (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑥, 𝑈} ∈ ran 𝐸))
110 prcom 4211 . . . . . . . . . . . . 13 {𝑥, 𝑈} = {𝑈, 𝑥}
111110eleq1i 2679 . . . . . . . . . . . 12 ({𝑥, 𝑈} ∈ ran 𝐸 ↔ {𝑈, 𝑥} ∈ ran 𝐸)
112109, 111syl6bb 275 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑈, 𝑥} ∈ ran 𝐸))
113112biimpd 218 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → (𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) → {𝑈, 𝑥} ∈ ran 𝐸))
114108, 113syl5bi 231 . . . . . . . . 9 (𝑉 USGrph 𝐸 → (𝑥𝑁 → {𝑈, 𝑥} ∈ ran 𝐸))
115114adantr 480 . . . . . . . 8 ((𝑉 USGrph 𝐸𝑈𝑉) → (𝑥𝑁 → {𝑈, 𝑥} ∈ ran 𝐸))
116115imp 444 . . . . . . 7 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑥𝑁) → {𝑈, 𝑥} ∈ ran 𝐸)
117116adantr 480 . . . . . 6 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → {𝑈, 𝑥} ∈ ran 𝐸)
1181eleq2i 2680 . . . . . . . . . 10 (𝑦𝑁𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈))
119 nbgraeledg 25959 . . . . . . . . . . . 12 (𝑉 USGrph 𝐸 → (𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑦, 𝑈} ∈ ran 𝐸))
120 prcom 4211 . . . . . . . . . . . . 13 {𝑦, 𝑈} = {𝑈, 𝑦}
121120eleq1i 2679 . . . . . . . . . . . 12 ({𝑦, 𝑈} ∈ ran 𝐸 ↔ {𝑈, 𝑦} ∈ ran 𝐸)
122119, 121syl6bb 275 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → (𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑈, 𝑦} ∈ ran 𝐸))
123122biimpd 218 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → (𝑦 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) → {𝑈, 𝑦} ∈ ran 𝐸))
124118, 123syl5bi 231 . . . . . . . . 9 (𝑉 USGrph 𝐸 → (𝑦𝑁 → {𝑈, 𝑦} ∈ ran 𝐸))
125124adantr 480 . . . . . . . 8 ((𝑉 USGrph 𝐸𝑈𝑉) → (𝑦𝑁 → {𝑈, 𝑦} ∈ ran 𝐸))
126125adantr 480 . . . . . . 7 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑥𝑁) → (𝑦𝑁 → {𝑈, 𝑦} ∈ ran 𝐸))
127126imp 444 . . . . . 6 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → {𝑈, 𝑦} ∈ ran 𝐸)
128 f1ocnvfvrneq 6441 . . . . . . 7 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ({𝑈, 𝑥} ∈ ran 𝐸 ∧ {𝑈, 𝑦} ∈ ran 𝐸)) → ((𝐸‘{𝑈, 𝑥}) = (𝐸‘{𝑈, 𝑦}) → {𝑈, 𝑥} = {𝑈, 𝑦}))
129 vex 3176 . . . . . . . 8 𝑥 ∈ V
130 vex 3176 . . . . . . . 8 𝑦 ∈ V
131129, 130preqr2 4321 . . . . . . 7 ({𝑈, 𝑥} = {𝑈, 𝑦} → 𝑥 = 𝑦)
132128, 131syl6 34 . . . . . 6 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ({𝑈, 𝑥} ∈ ran 𝐸 ∧ {𝑈, 𝑦} ∈ ran 𝐸)) → ((𝐸‘{𝑈, 𝑥}) = (𝐸‘{𝑈, 𝑦}) → 𝑥 = 𝑦))
133107, 117, 127, 132syl12anc 1316 . . . . 5 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → ((𝐸‘{𝑈, 𝑥}) = (𝐸‘{𝑈, 𝑦}) → 𝑥 = 𝑦))
134105, 133sylbid 229 . . . 4 ((((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑥𝑁) ∧ 𝑦𝑁) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
135134ralrimiva 2949 . . 3 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑥𝑁) → ∀𝑦𝑁 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
136135ralrimiva 2949 . 2 ((𝑉 USGrph 𝐸𝑈𝑉) → ∀𝑥𝑁𝑦𝑁 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
137 dff1o6 6431 . 2 (𝐹:𝑁1-1-onto𝐼 ↔ (𝐹 Fn 𝑁 ∧ ran 𝐹 = 𝐼 ∧ ∀𝑥𝑁𝑦𝑁 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
1386, 96, 136, 137syl3anbrc 1239 1 ((𝑉 USGrph 𝐸𝑈𝑉) → 𝐹:𝑁1-1-onto𝐼)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  {crab 2900  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  ℩crio 6510  (class class class)co 6549   USGrph cusg 25859   Neighbors cnbgra 25946 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862  df-nbgra 25949 This theorem is referenced by:  nbgraf1o0  25975
 Copyright terms: Public domain W3C validator