Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frgrncvvdeqlemC Structured version   Visualization version   GIF version

Theorem frgrncvvdeqlemC 41478
 Description: Lemma C for frgrncvvdeq 41480. This corresponds to statement 3 in [Huneke] p. 1: "By symmetry the map is onto". (Contributed by Alexander van der Vekens, 24-Dec-2017.) (Revised by AV, 10-May-2021.)
Hypotheses
Ref Expression
frgrncvvdeq.v1 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.e 𝐸 = (Edg‘𝐺)
frgrncvvdeq.nx 𝐷 = (𝐺 NeighbVtx 𝑋)
frgrncvvdeq.ny 𝑁 = (𝐺 NeighbVtx 𝑌)
frgrncvvdeq.x (𝜑𝑋𝑉)
frgrncvvdeq.y (𝜑𝑌𝑉)
frgrncvvdeq.ne (𝜑𝑋𝑌)
frgrncvvdeq.xy (𝜑𝑌𝐷)
frgrncvvdeq.f (𝜑𝐺 ∈ FriendGraph )
frgrncvvdeq.a 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
Assertion
Ref Expression
frgrncvvdeqlemC (𝜑𝐴:𝐷onto𝑁)
Distinct variable groups:   𝑦,𝐷   𝑦,𝐺   𝑦,𝑉   𝑦,𝑌   𝜑,𝑦,𝑥   𝑦,𝐸   𝑦,𝑁   𝑥,𝐷   𝑥,𝑁   𝜑,𝑥   𝑥,𝐸
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐺(𝑥)   𝑉(𝑥)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem frgrncvvdeqlemC
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrncvvdeq.v1 . . 3 𝑉 = (Vtx‘𝐺)
2 frgrncvvdeq.e . . 3 𝐸 = (Edg‘𝐺)
3 frgrncvvdeq.nx . . 3 𝐷 = (𝐺 NeighbVtx 𝑋)
4 frgrncvvdeq.ny . . 3 𝑁 = (𝐺 NeighbVtx 𝑌)
5 frgrncvvdeq.x . . 3 (𝜑𝑋𝑉)
6 frgrncvvdeq.y . . 3 (𝜑𝑌𝑉)
7 frgrncvvdeq.ne . . 3 (𝜑𝑋𝑌)
8 frgrncvvdeq.xy . . 3 (𝜑𝑌𝐷)
9 frgrncvvdeq.f . . 3 (𝜑𝐺 ∈ FriendGraph )
10 frgrncvvdeq.a . . 3 𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10frgrncvvdeqlem5 41473 . 2 (𝜑𝐴:𝐷𝑁)
129adantr 480 . . . . . . 7 ((𝜑𝑛𝑁) → 𝐺 ∈ FriendGraph )
134eleq2i 2680 . . . . . . . . . 10 (𝑛𝑁𝑛 ∈ (𝐺 NeighbVtx 𝑌))
14 frgrusgr 41432 . . . . . . . . . . 11 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
151nbgrisvtx 40581 . . . . . . . . . . . 12 ((𝐺 ∈ USGraph ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) → 𝑛𝑉)
1615ex 449 . . . . . . . . . . 11 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛𝑉))
179, 14, 163syl 18 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛𝑉))
1813, 17syl5bi 231 . . . . . . . . 9 (𝜑 → (𝑛𝑁𝑛𝑉))
1918imp 444 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑛𝑉)
205adantr 480 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑋𝑉)
211, 2, 3, 4, 5, 6, 7, 8, 9, 10frgrncvvdeqlem2 41470 . . . . . . . . . 10 (𝜑𝑋𝑁)
22 df-nel 2783 . . . . . . . . . . 11 (𝑋𝑁 ↔ ¬ 𝑋𝑁)
23 nelelne 2880 . . . . . . . . . . 11 𝑋𝑁 → (𝑛𝑁𝑛𝑋))
2422, 23sylbi 206 . . . . . . . . . 10 (𝑋𝑁 → (𝑛𝑁𝑛𝑋))
2521, 24syl 17 . . . . . . . . 9 (𝜑 → (𝑛𝑁𝑛𝑋))
2625imp 444 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑛𝑋)
2719, 20, 263jca 1235 . . . . . . 7 ((𝜑𝑛𝑁) → (𝑛𝑉𝑋𝑉𝑛𝑋))
2812, 27jca 553 . . . . . 6 ((𝜑𝑛𝑁) → (𝐺 ∈ FriendGraph ∧ (𝑛𝑉𝑋𝑉𝑛𝑋)))
291, 2frcond2 41439 . . . . . . 7 (𝐺 ∈ FriendGraph → ((𝑛𝑉𝑋𝑉𝑛𝑋) → ∃!𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
3029imp 444 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ (𝑛𝑉𝑋𝑉𝑛𝑋)) → ∃!𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))
31 reurex 3137 . . . . . . 7 (∃!𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → ∃𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))
32 df-rex 2902 . . . . . . 7 (∃𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) ↔ ∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
3331, 32sylib 207 . . . . . 6 (∃!𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → ∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
3428, 30, 333syl 18 . . . . 5 ((𝜑𝑛𝑁) → ∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
359, 14syl 17 . . . . . . . 8 (𝜑𝐺 ∈ USGraph )
36 simprrr 801 . . . . . . . . . . 11 ((((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))) → {𝑚, 𝑋} ∈ 𝐸)
373eleq2i 2680 . . . . . . . . . . . 12 (𝑚𝐷𝑚 ∈ (𝐺 NeighbVtx 𝑋))
382nbusgreledg 40575 . . . . . . . . . . . . . 14 (𝐺 ∈ USGraph → (𝑚 ∈ (𝐺 NeighbVtx 𝑋) ↔ {𝑚, 𝑋} ∈ 𝐸))
3938ad2antlr 759 . . . . . . . . . . . . 13 (((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) → (𝑚 ∈ (𝐺 NeighbVtx 𝑋) ↔ {𝑚, 𝑋} ∈ 𝐸))
4039adantr 480 . . . . . . . . . . . 12 ((((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))) → (𝑚 ∈ (𝐺 NeighbVtx 𝑋) ↔ {𝑚, 𝑋} ∈ 𝐸))
4137, 40syl5bb 271 . . . . . . . . . . 11 ((((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))) → (𝑚𝐷 ↔ {𝑚, 𝑋} ∈ 𝐸))
4236, 41mpbird 246 . . . . . . . . . 10 ((((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))) → 𝑚𝐷)
432nbusgreledg 40575 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) ↔ {𝑛, 𝑚} ∈ 𝐸))
4443biimprcd 239 . . . . . . . . . . . . . . . 16 ({𝑛, 𝑚} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
4544adantr 480 . . . . . . . . . . . . . . 15 (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝐺 ∈ USGraph → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
4645adantl 481 . . . . . . . . . . . . . 14 ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝐺 ∈ USGraph → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
4746com12 32 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
4847ad2antlr 759 . . . . . . . . . . . 12 (((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
4948imp 444 . . . . . . . . . . 11 ((((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚))
50 elin 3758 . . . . . . . . . . . . . . 15 (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ (𝑛 ∈ (𝐺 NeighbVtx 𝑚) ∧ 𝑛𝑁))
51 simpll 786 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝐺 ∈ USGraph ) ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝜑)
5238bicomd 212 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 ∈ USGraph → ({𝑚, 𝑋} ∈ 𝐸𝑚 ∈ (𝐺 NeighbVtx 𝑋)))
5352adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝐺 ∈ USGraph ) → ({𝑚, 𝑋} ∈ 𝐸𝑚 ∈ (𝐺 NeighbVtx 𝑋)))
5453biimpa 500 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝐺 ∈ USGraph ) ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑚 ∈ (𝐺 NeighbVtx 𝑋))
5554, 37sylibr 223 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝐺 ∈ USGraph ) ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑚𝐷)
5651, 55jca 553 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝐺 ∈ USGraph ) ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝜑𝑚𝐷))
57 preq1 4212 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑚 → {𝑥, 𝑦} = {𝑚, 𝑦})
5857eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑚 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑚, 𝑦} ∈ 𝐸))
5958riotabidv 6513 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑚 → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = (𝑦𝑁 {𝑚, 𝑦} ∈ 𝐸))
6059cbvmptv 4678 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑚𝐷 ↦ (𝑦𝑁 {𝑚, 𝑦} ∈ 𝐸))
6110, 60eqtri 2632 . . . . . . . . . . . . . . . . . . . 20 𝐴 = (𝑚𝐷 ↦ (𝑦𝑁 {𝑚, 𝑦} ∈ 𝐸))
621, 2, 3, 4, 5, 6, 7, 8, 9, 61frgrncvvdeqlem6 41474 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚𝐷) → {(𝐴𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁))
63 eleq2 2677 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 NeighbVtx 𝑚) ∩ 𝑁) = {(𝐴𝑚)} → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴𝑚)}))
6463eqcoms 2618 . . . . . . . . . . . . . . . . . . . 20 ({(𝐴𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴𝑚)}))
65 elsni 4142 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ {(𝐴𝑚)} → 𝑛 = (𝐴𝑚))
6664, 65syl6bi 242 . . . . . . . . . . . . . . . . . . 19 ({(𝐴𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → 𝑛 = (𝐴𝑚)))
6756, 62, 663syl 18 . . . . . . . . . . . . . . . . . 18 (((𝜑𝐺 ∈ USGraph ) ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → 𝑛 = (𝐴𝑚)))
6867expcom 450 . . . . . . . . . . . . . . . . 17 ({𝑚, 𝑋} ∈ 𝐸 → ((𝜑𝐺 ∈ USGraph ) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → 𝑛 = (𝐴𝑚))))
6968ad2antll 761 . . . . . . . . . . . . . . . 16 ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → ((𝜑𝐺 ∈ USGraph ) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → 𝑛 = (𝐴𝑚))))
7069com3r 85 . . . . . . . . . . . . . . 15 (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → ((𝜑𝐺 ∈ USGraph ) → 𝑛 = (𝐴𝑚))))
7150, 70sylbir 224 . . . . . . . . . . . . . 14 ((𝑛 ∈ (𝐺 NeighbVtx 𝑚) ∧ 𝑛𝑁) → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → ((𝜑𝐺 ∈ USGraph ) → 𝑛 = (𝐴𝑚))))
7271ex 449 . . . . . . . . . . . . 13 (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → (𝑛𝑁 → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → ((𝜑𝐺 ∈ USGraph ) → 𝑛 = (𝐴𝑚)))))
7372com14 94 . . . . . . . . . . . 12 ((𝜑𝐺 ∈ USGraph ) → (𝑛𝑁 → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → 𝑛 = (𝐴𝑚)))))
7473imp31 447 . . . . . . . . . . 11 ((((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))) → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → 𝑛 = (𝐴𝑚)))
7549, 74mpd 15 . . . . . . . . . 10 ((((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))) → 𝑛 = (𝐴𝑚))
7642, 75jca 553 . . . . . . . . 9 ((((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))) → (𝑚𝐷𝑛 = (𝐴𝑚)))
7776exp31 628 . . . . . . . 8 ((𝜑𝐺 ∈ USGraph ) → (𝑛𝑁 → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑚𝐷𝑛 = (𝐴𝑚)))))
7835, 77mpdan 699 . . . . . . 7 (𝜑 → (𝑛𝑁 → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑚𝐷𝑛 = (𝐴𝑚)))))
7978imp 444 . . . . . 6 ((𝜑𝑛𝑁) → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑚𝐷𝑛 = (𝐴𝑚))))
8079eximdv 1833 . . . . 5 ((𝜑𝑛𝑁) → (∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚))))
8134, 80mpd 15 . . . 4 ((𝜑𝑛𝑁) → ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚)))
82 df-rex 2902 . . . 4 (∃𝑚𝐷 𝑛 = (𝐴𝑚) ↔ ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚)))
8381, 82sylibr 223 . . 3 ((𝜑𝑛𝑁) → ∃𝑚𝐷 𝑛 = (𝐴𝑚))
8483ralrimiva 2949 . 2 (𝜑 → ∀𝑛𝑁𝑚𝐷 𝑛 = (𝐴𝑚))
85 dffo3 6282 . 2 (𝐴:𝐷onto𝑁 ↔ (𝐴:𝐷𝑁 ∧ ∀𝑛𝑁𝑚𝐷 𝑛 = (𝐴𝑚)))
8611, 84, 85sylanbrc 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   ↦ cmpt 4643  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   NeighbVtx cnbgr 40550   FriendGraph cfrgr 41428 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-fal 1481  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-upgr 25749  df-umgr 25750  df-edga 25793  df-usgr 40381  df-nbgr 40554  df-frgr 41429 This theorem is referenced by:  frgrncvvdeqlem8  41479
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