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

Theorem frgrancvvdeqlemC 26566
 Description: Lemma C for frgrancvvdeq 26569. This corresponds to statement 3 in [Huneke] p. 1: "By symmetry the map is onto". (Contributed by Alexander van der Vekens, 24-Dec-2017.)
Hypotheses
Ref Expression
frgrancvvdeq.nx 𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)
frgrancvvdeq.ny 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)
frgrancvvdeq.x (𝜑𝑋𝑉)
frgrancvvdeq.y (𝜑𝑌𝑉)
frgrancvvdeq.ne (𝜑𝑋𝑌)
frgrancvvdeq.xy (𝜑𝑌𝐷)
frgrancvvdeq.f (𝜑𝑉 FriendGrph 𝐸)
frgrancvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))
Assertion
Ref Expression
frgrancvvdeqlemC (𝜑𝐴:𝐷onto𝑁)
Distinct variable groups:   𝑦,𝐷,𝑥   𝑥,𝑉,𝑦   𝑥,𝐸,𝑦   𝑦,𝑌   𝜑,𝑦   𝑦,𝑁   𝑥,𝐷   𝑥,𝑁   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrancvvdeqlemC
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrancvvdeq.nx . . 3 𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)
2 frgrancvvdeq.ny . . 3 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)
3 frgrancvvdeq.x . . 3 (𝜑𝑋𝑉)
4 frgrancvvdeq.y . . 3 (𝜑𝑌𝑉)
5 frgrancvvdeq.ne . . 3 (𝜑𝑋𝑌)
6 frgrancvvdeq.xy . . 3 (𝜑𝑌𝐷)
7 frgrancvvdeq.f . . 3 (𝜑𝑉 FriendGrph 𝐸)
8 frgrancvvdeq.a . . 3 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))
91, 2, 3, 4, 5, 6, 7, 8frgrancvvdeqlem5 26561 . 2 (𝜑𝐴:𝐷𝑁)
107adantr 480 . . . . . . 7 ((𝜑𝑛𝑁) → 𝑉 FriendGrph 𝐸)
112eleq2i 2680 . . . . . . . . . 10 (𝑛𝑁𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑌))
12 frisusgra 26519 . . . . . . . . . . 11 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
13 nbgraisvtx 25960 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑌) → 𝑛𝑉))
147, 12, 133syl 18 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑌) → 𝑛𝑉))
1511, 14syl5bi 231 . . . . . . . . 9 (𝜑 → (𝑛𝑁𝑛𝑉))
1615imp 444 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑛𝑉)
173adantr 480 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑋𝑉)
181, 2, 3, 4, 5, 6, 7, 8frgrancvvdeqlem2 26558 . . . . . . . . . 10 (𝜑𝑋𝑁)
19 df-nel 2783 . . . . . . . . . . 11 (𝑋𝑁 ↔ ¬ 𝑋𝑁)
20 eleq1 2676 . . . . . . . . . . . . . 14 (𝑛 = 𝑋 → (𝑛𝑁𝑋𝑁))
2120biimpcd 238 . . . . . . . . . . . . 13 (𝑛𝑁 → (𝑛 = 𝑋𝑋𝑁))
2221con3rr3 150 . . . . . . . . . . . 12 𝑋𝑁 → (𝑛𝑁 → ¬ 𝑛 = 𝑋))
23 df-ne 2782 . . . . . . . . . . . 12 (𝑛𝑋 ↔ ¬ 𝑛 = 𝑋)
2422, 23syl6ibr 241 . . . . . . . . . . 11 𝑋𝑁 → (𝑛𝑁𝑛𝑋))
2519, 24sylbi 206 . . . . . . . . . 10 (𝑋𝑁 → (𝑛𝑁𝑛𝑋))
2618, 25syl 17 . . . . . . . . 9 (𝜑 → (𝑛𝑁𝑛𝑋))
2726imp 444 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑛𝑋)
2816, 17, 273jca 1235 . . . . . . 7 ((𝜑𝑛𝑁) → (𝑛𝑉𝑋𝑉𝑛𝑋))
2910, 28jca 553 . . . . . 6 ((𝜑𝑛𝑁) → (𝑉 FriendGrph 𝐸 ∧ (𝑛𝑉𝑋𝑉𝑛𝑋)))
30 frgraun 26523 . . . . . . 7 (𝑉 FriendGrph 𝐸 → ((𝑛𝑉𝑋𝑉𝑛𝑋) → ∃!𝑚𝑉 ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)))
3130imp 444 . . . . . 6 ((𝑉 FriendGrph 𝐸 ∧ (𝑛𝑉𝑋𝑉𝑛𝑋)) → ∃!𝑚𝑉 ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸))
32 reurex 3137 . . . . . . 7 (∃!𝑚𝑉 ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸) → ∃𝑚𝑉 ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸))
33 df-rex 2902 . . . . . . 7 (∃𝑚𝑉 ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸) ↔ ∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)))
3432, 33sylib 207 . . . . . 6 (∃!𝑚𝑉 ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸) → ∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)))
3529, 31, 343syl 18 . . . . 5 ((𝜑𝑛𝑁) → ∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)))
367, 12syl 17 . . . . . . . 8 (𝜑𝑉 USGrph 𝐸)
37 simprrr 801 . . . . . . . . . . . 12 ((((𝜑𝑉 USGrph 𝐸) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸))) → {𝑚, 𝑋} ∈ ran 𝐸)
381eleq2i 2680 . . . . . . . . . . . . 13 (𝑚𝐷𝑚 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))
39 nbgraeledg 25959 . . . . . . . . . . . . . . 15 (𝑉 USGrph 𝐸 → (𝑚 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ↔ {𝑚, 𝑋} ∈ ran 𝐸))
4039ad2antlr 759 . . . . . . . . . . . . . 14 (((𝜑𝑉 USGrph 𝐸) ∧ 𝑛𝑁) → (𝑚 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ↔ {𝑚, 𝑋} ∈ ran 𝐸))
4140adantr 480 . . . . . . . . . . . . 13 ((((𝜑𝑉 USGrph 𝐸) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸))) → (𝑚 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ↔ {𝑚, 𝑋} ∈ ran 𝐸))
4238, 41syl5bb 271 . . . . . . . . . . . 12 ((((𝜑𝑉 USGrph 𝐸) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸))) → (𝑚𝐷 ↔ {𝑚, 𝑋} ∈ ran 𝐸))
4337, 42mpbird 246 . . . . . . . . . . 11 ((((𝜑𝑉 USGrph 𝐸) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸))) → 𝑚𝐷)
44 nbgraeledg 25959 . . . . . . . . . . . . . . . . . 18 (𝑉 USGrph 𝐸 → (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑚) ↔ {𝑛, 𝑚} ∈ ran 𝐸))
4544biimprcd 239 . . . . . . . . . . . . . . . . 17 ({𝑛, 𝑚} ∈ ran 𝐸 → (𝑉 USGrph 𝐸𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑚)))
4645adantr 480 . . . . . . . . . . . . . . . 16 (({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸) → (𝑉 USGrph 𝐸𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑚)))
4746adantl 481 . . . . . . . . . . . . . . 15 ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)) → (𝑉 USGrph 𝐸𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑚)))
4847com12 32 . . . . . . . . . . . . . 14 (𝑉 USGrph 𝐸 → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)) → 𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑚)))
4948ad2antlr 759 . . . . . . . . . . . . 13 (((𝜑𝑉 USGrph 𝐸) ∧ 𝑛𝑁) → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)) → 𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑚)))
5049imp 444 . . . . . . . . . . . 12 ((((𝜑𝑉 USGrph 𝐸) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸))) → 𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑚))
51 elin 3758 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑚) ∩ 𝑁) ↔ (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑚) ∧ 𝑛𝑁))
52 simpll 786 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑉 USGrph 𝐸) ∧ {𝑚, 𝑋} ∈ ran 𝐸) → 𝜑)
5339bicomd 212 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑉 USGrph 𝐸 → ({𝑚, 𝑋} ∈ ran 𝐸𝑚 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))
5453adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑉 USGrph 𝐸) → ({𝑚, 𝑋} ∈ ran 𝐸𝑚 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))
5554biimpa 500 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑉 USGrph 𝐸) ∧ {𝑚, 𝑋} ∈ ran 𝐸) → 𝑚 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))
5655, 38sylibr 223 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑉 USGrph 𝐸) ∧ {𝑚, 𝑋} ∈ ran 𝐸) → 𝑚𝐷)
5752, 56jca 553 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑉 USGrph 𝐸) ∧ {𝑚, 𝑋} ∈ ran 𝐸) → (𝜑𝑚𝐷))
58 preq1 4212 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑚 → {𝑥, 𝑦} = {𝑚, 𝑦})
5958eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑚 → ({𝑥, 𝑦} ∈ ran 𝐸 ↔ {𝑚, 𝑦} ∈ ran 𝐸))
6059riotabidv 6513 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑚 → (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸) = (𝑦𝑁 {𝑚, 𝑦} ∈ ran 𝐸))
6160cbvmptv 4678 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸)) = (𝑚𝐷 ↦ (𝑦𝑁 {𝑚, 𝑦} ∈ ran 𝐸))
628, 61eqtri 2632 . . . . . . . . . . . . . . . . . . . . 21 𝐴 = (𝑚𝐷 ↦ (𝑦𝑁 {𝑚, 𝑦} ∈ ran 𝐸))
631, 2, 3, 4, 5, 6, 7, 62frgrancvvdeqlem6 26562 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚𝐷) → {(𝐴𝑚)} = ((⟨𝑉, 𝐸⟩ Neighbors 𝑚) ∩ 𝑁))
64 eleq2 2677 . . . . . . . . . . . . . . . . . . . . . 22 (((⟨𝑉, 𝐸⟩ Neighbors 𝑚) ∩ 𝑁) = {(𝐴𝑚)} → (𝑛 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴𝑚)}))
6564eqcoms 2618 . . . . . . . . . . . . . . . . . . . . 21 ({(𝐴𝑚)} = ((⟨𝑉, 𝐸⟩ Neighbors 𝑚) ∩ 𝑁) → (𝑛 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴𝑚)}))
66 elsni 4142 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ {(𝐴𝑚)} → 𝑛 = (𝐴𝑚))
6765, 66syl6bi 242 . . . . . . . . . . . . . . . . . . . 20 ({(𝐴𝑚)} = ((⟨𝑉, 𝐸⟩ Neighbors 𝑚) ∩ 𝑁) → (𝑛 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑚) ∩ 𝑁) → 𝑛 = (𝐴𝑚)))
6857, 63, 673syl 18 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑉 USGrph 𝐸) ∧ {𝑚, 𝑋} ∈ ran 𝐸) → (𝑛 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑚) ∩ 𝑁) → 𝑛 = (𝐴𝑚)))
6968expcom 450 . . . . . . . . . . . . . . . . . 18 ({𝑚, 𝑋} ∈ ran 𝐸 → ((𝜑𝑉 USGrph 𝐸) → (𝑛 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑚) ∩ 𝑁) → 𝑛 = (𝐴𝑚))))
7069ad2antll 761 . . . . . . . . . . . . . . . . 17 ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)) → ((𝜑𝑉 USGrph 𝐸) → (𝑛 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑚) ∩ 𝑁) → 𝑛 = (𝐴𝑚))))
7170com3r 85 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑚) ∩ 𝑁) → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)) → ((𝜑𝑉 USGrph 𝐸) → 𝑛 = (𝐴𝑚))))
7251, 71sylbir 224 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑚) ∧ 𝑛𝑁) → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)) → ((𝜑𝑉 USGrph 𝐸) → 𝑛 = (𝐴𝑚))))
7372ex 449 . . . . . . . . . . . . . 14 (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑚) → (𝑛𝑁 → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)) → ((𝜑𝑉 USGrph 𝐸) → 𝑛 = (𝐴𝑚)))))
7473com14 94 . . . . . . . . . . . . 13 ((𝜑𝑉 USGrph 𝐸) → (𝑛𝑁 → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)) → (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑚) → 𝑛 = (𝐴𝑚)))))
7574imp31 447 . . . . . . . . . . . 12 ((((𝜑𝑉 USGrph 𝐸) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸))) → (𝑛 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑚) → 𝑛 = (𝐴𝑚)))
7650, 75mpd 15 . . . . . . . . . . 11 ((((𝜑𝑉 USGrph 𝐸) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸))) → 𝑛 = (𝐴𝑚))
7743, 76jca 553 . . . . . . . . . 10 ((((𝜑𝑉 USGrph 𝐸) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸))) → (𝑚𝐷𝑛 = (𝐴𝑚)))
7877ex 449 . . . . . . . . 9 (((𝜑𝑉 USGrph 𝐸) ∧ 𝑛𝑁) → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)) → (𝑚𝐷𝑛 = (𝐴𝑚))))
7978ex 449 . . . . . . . 8 ((𝜑𝑉 USGrph 𝐸) → (𝑛𝑁 → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)) → (𝑚𝐷𝑛 = (𝐴𝑚)))))
8036, 79mpdan 699 . . . . . . 7 (𝜑 → (𝑛𝑁 → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)) → (𝑚𝐷𝑛 = (𝐴𝑚)))))
8180imp 444 . . . . . 6 ((𝜑𝑛𝑁) → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)) → (𝑚𝐷𝑛 = (𝐴𝑚))))
8281eximdv 1833 . . . . 5 ((𝜑𝑛𝑁) → (∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ ran 𝐸 ∧ {𝑚, 𝑋} ∈ ran 𝐸)) → ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚))))
8335, 82mpd 15 . . . 4 ((𝜑𝑛𝑁) → ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚)))
84 df-rex 2902 . . . 4 (∃𝑚𝐷 𝑛 = (𝐴𝑚) ↔ ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚)))
8583, 84sylibr 223 . . 3 ((𝜑𝑛𝑁) → ∃𝑚𝐷 𝑛 = (𝐴𝑚))
8685ralrimiva 2949 . 2 (𝜑 → ∀𝑛𝑁𝑚𝐷 𝑛 = (𝐴𝑚))
87 dffo3 6282 . 2 (𝐴:𝐷onto𝑁 ↔ (𝐴:𝐷𝑁 ∧ ∀𝑛𝑁𝑚𝐷 𝑛 = (𝐴𝑚)))
889, 86, 87sylanbrc 695 1 (𝜑𝐴:𝐷onto𝑁)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898   ∩ cin 3539  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  ℩crio 6510  (class class class)co 6549   USGrph cusg 25859   Neighbors cnbgra 25946   FriendGrph cfrgra 26515 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-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-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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  df-frgra 26516 This theorem is referenced by:  frgrancvvdeqlem8  26567
 Copyright terms: Public domain W3C validator