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Theorem frgrncvvdeqlemC 27070
 Description: Lemma C for frgrncvvdeq 27072. 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 27065 . 2 (𝜑𝐴:𝐷𝑁)
129adantr 481 . . . . . . 7 ((𝜑𝑛𝑁) → 𝐺 ∈ FriendGraph )
134eleq2i 2690 . . . . . . . . . 10 (𝑛𝑁𝑛 ∈ (𝐺 NeighbVtx 𝑌))
14 frgrusgr 27024 . . . . . . . . . . 11 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
151nbgrisvtx 26176 . . . . . . . . . . . 12 ((𝐺 ∈ USGraph ∧ 𝑛 ∈ (𝐺 NeighbVtx 𝑌)) → 𝑛𝑉)
1615ex 450 . . . . . . . . . . 11 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛𝑉))
179, 14, 163syl 18 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ (𝐺 NeighbVtx 𝑌) → 𝑛𝑉))
1813, 17syl5bi 232 . . . . . . . . 9 (𝜑 → (𝑛𝑁𝑛𝑉))
1918imp 445 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑛𝑉)
205adantr 481 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑋𝑉)
211, 2, 3, 4, 5, 6, 7, 8, 9, 10frgrncvvdeqlem2 27062 . . . . . . . . . 10 (𝜑𝑋𝑁)
22 df-nel 2894 . . . . . . . . . . 11 (𝑋𝑁 ↔ ¬ 𝑋𝑁)
23 nelelne 2888 . . . . . . . . . . 11 𝑋𝑁 → (𝑛𝑁𝑛𝑋))
2422, 23sylbi 207 . . . . . . . . . 10 (𝑋𝑁 → (𝑛𝑁𝑛𝑋))
2521, 24syl 17 . . . . . . . . 9 (𝜑 → (𝑛𝑁𝑛𝑋))
2625imp 445 . . . . . . . 8 ((𝜑𝑛𝑁) → 𝑛𝑋)
2719, 20, 263jca 1240 . . . . . . 7 ((𝜑𝑛𝑁) → (𝑛𝑉𝑋𝑉𝑛𝑋))
2812, 27jca 554 . . . . . 6 ((𝜑𝑛𝑁) → (𝐺 ∈ FriendGraph ∧ (𝑛𝑉𝑋𝑉𝑛𝑋)))
291, 2frcond2 27031 . . . . . . 7 (𝐺 ∈ FriendGraph → ((𝑛𝑉𝑋𝑉𝑛𝑋) → ∃!𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
3029imp 445 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ (𝑛𝑉𝑋𝑉𝑛𝑋)) → ∃!𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))
31 reurex 3153 . . . . . . 7 (∃!𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → ∃𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))
32 df-rex 2914 . . . . . . 7 (∃𝑚𝑉 ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) ↔ ∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)))
3331, 32sylib 208 . . . . . 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 804 . . . . . . . . . . 11 ((((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))) → {𝑚, 𝑋} ∈ 𝐸)
373eleq2i 2690 . . . . . . . . . . . 12 (𝑚𝐷𝑚 ∈ (𝐺 NeighbVtx 𝑋))
382nbusgreledg 26170 . . . . . . . . . . . . . 14 (𝐺 ∈ USGraph → (𝑚 ∈ (𝐺 NeighbVtx 𝑋) ↔ {𝑚, 𝑋} ∈ 𝐸))
3938ad2antlr 762 . . . . . . . . . . . . 13 (((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) → (𝑚 ∈ (𝐺 NeighbVtx 𝑋) ↔ {𝑚, 𝑋} ∈ 𝐸))
4039adantr 481 . . . . . . . . . . . 12 ((((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))) → (𝑚 ∈ (𝐺 NeighbVtx 𝑋) ↔ {𝑚, 𝑋} ∈ 𝐸))
4137, 40syl5bb 272 . . . . . . . . . . 11 ((((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))) → (𝑚𝐷 ↔ {𝑚, 𝑋} ∈ 𝐸))
4236, 41mpbird 247 . . . . . . . . . 10 ((((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))) → 𝑚𝐷)
432nbusgreledg 26170 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ USGraph → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) ↔ {𝑛, 𝑚} ∈ 𝐸))
4443biimprcd 240 . . . . . . . . . . . . . . . 16 ({𝑛, 𝑚} ∈ 𝐸 → (𝐺 ∈ USGraph → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
4544adantr 481 . . . . . . . . . . . . . . 15 (({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝐺 ∈ USGraph → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
4645adantl 482 . . . . . . . . . . . . . 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 762 . . . . . . . . . . . 12 (((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚)))
4948imp 445 . . . . . . . . . . 11 ((((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))) → 𝑛 ∈ (𝐺 NeighbVtx 𝑚))
50 elin 3780 . . . . . . . . . . . . . . 15 (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ (𝑛 ∈ (𝐺 NeighbVtx 𝑚) ∧ 𝑛𝑁))
51 simpll 789 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝐺 ∈ USGraph ) ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝜑)
5238bicomd 213 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐺 ∈ USGraph → ({𝑚, 𝑋} ∈ 𝐸𝑚 ∈ (𝐺 NeighbVtx 𝑋)))
5352adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝐺 ∈ USGraph ) → ({𝑚, 𝑋} ∈ 𝐸𝑚 ∈ (𝐺 NeighbVtx 𝑋)))
5453biimpa 501 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝐺 ∈ USGraph ) ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑚 ∈ (𝐺 NeighbVtx 𝑋))
5554, 37sylibr 224 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝐺 ∈ USGraph ) ∧ {𝑚, 𝑋} ∈ 𝐸) → 𝑚𝐷)
5651, 55jca 554 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝐺 ∈ USGraph ) ∧ {𝑚, 𝑋} ∈ 𝐸) → (𝜑𝑚𝐷))
57 preq1 4245 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑚 → {𝑥, 𝑦} = {𝑚, 𝑦})
5857eleq1d 2683 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑚 → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑚, 𝑦} ∈ 𝐸))
5958riotabidv 6578 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑚 → (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸) = (𝑦𝑁 {𝑚, 𝑦} ∈ 𝐸))
6059cbvmptv 4720 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ 𝐸)) = (𝑚𝐷 ↦ (𝑦𝑁 {𝑚, 𝑦} ∈ 𝐸))
6110, 60eqtri 2643 . . . . . . . . . . . . . . . . . . . 20 𝐴 = (𝑚𝐷 ↦ (𝑦𝑁 {𝑚, 𝑦} ∈ 𝐸))
621, 2, 3, 4, 5, 6, 7, 8, 9, 61frgrncvvdeqlem6 27066 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚𝐷) → {(𝐴𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁))
63 eleq2 2687 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 NeighbVtx 𝑚) ∩ 𝑁) = {(𝐴𝑚)} → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴𝑚)}))
6463eqcoms 2629 . . . . . . . . . . . . . . . . . . . 20 ({(𝐴𝑚)} = ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) ↔ 𝑛 ∈ {(𝐴𝑚)}))
65 elsni 4172 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ {(𝐴𝑚)} → 𝑛 = (𝐴𝑚))
6664, 65syl6bi 243 . . . . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . . 17 ({𝑚, 𝑋} ∈ 𝐸 → ((𝜑𝐺 ∈ USGraph ) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → 𝑛 = (𝐴𝑚))))
6968ad2antll 764 . . . . . . . . . . . . . . . 16 ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → ((𝜑𝐺 ∈ USGraph ) → (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → 𝑛 = (𝐴𝑚))))
7069com3r 87 . . . . . . . . . . . . . . 15 (𝑛 ∈ ((𝐺 NeighbVtx 𝑚) ∩ 𝑁) → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → ((𝜑𝐺 ∈ USGraph ) → 𝑛 = (𝐴𝑚))))
7150, 70sylbir 225 . . . . . . . . . . . . . 14 ((𝑛 ∈ (𝐺 NeighbVtx 𝑚) ∧ 𝑛𝑁) → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → ((𝜑𝐺 ∈ USGraph ) → 𝑛 = (𝐴𝑚))))
7271ex 450 . . . . . . . . . . . . 13 (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → (𝑛𝑁 → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → ((𝜑𝐺 ∈ USGraph ) → 𝑛 = (𝐴𝑚)))))
7372com14 96 . . . . . . . . . . . 12 ((𝜑𝐺 ∈ USGraph ) → (𝑛𝑁 → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑛 ∈ (𝐺 NeighbVtx 𝑚) → 𝑛 = (𝐴𝑚)))))
7473imp31 448 . . . . . . . . . . 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 554 . . . . . . . . 9 ((((𝜑𝐺 ∈ USGraph ) ∧ 𝑛𝑁) ∧ (𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸))) → (𝑚𝐷𝑛 = (𝐴𝑚)))
7776exp31 629 . . . . . . . 8 ((𝜑𝐺 ∈ USGraph ) → (𝑛𝑁 → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑚𝐷𝑛 = (𝐴𝑚)))))
7835, 77mpdan 701 . . . . . . 7 (𝜑 → (𝑛𝑁 → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑚𝐷𝑛 = (𝐴𝑚)))))
7978imp 445 . . . . . 6 ((𝜑𝑛𝑁) → ((𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → (𝑚𝐷𝑛 = (𝐴𝑚))))
8079eximdv 1843 . . . . 5 ((𝜑𝑛𝑁) → (∃𝑚(𝑚𝑉 ∧ ({𝑛, 𝑚} ∈ 𝐸 ∧ {𝑚, 𝑋} ∈ 𝐸)) → ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚))))
8134, 80mpd 15 . . . 4 ((𝜑𝑛𝑁) → ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚)))
82 df-rex 2914 . . . 4 (∃𝑚𝐷 𝑛 = (𝐴𝑚) ↔ ∃𝑚(𝑚𝐷𝑛 = (𝐴𝑚)))
8381, 82sylibr 224 . . 3 ((𝜑𝑛𝑁) → ∃𝑚𝐷 𝑛 = (𝐴𝑚))
8483ralrimiva 2962 . 2 (𝜑 → ∀𝑛𝑁𝑚𝐷 𝑛 = (𝐴𝑚))
85 dffo3 6340 . 2 (𝐴:𝐷onto𝑁 ↔ (𝐴:𝐷𝑁 ∧ ∀𝑛𝑁𝑚𝐷 𝑛 = (𝐴𝑚)))
8611, 84, 85sylanbrc 697 1 (𝜑𝐴:𝐷onto𝑁)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987   ≠ wne 2790   ∉ wnel 2893  ∀wral 2908  ∃wrex 2909  ∃!wreu 2910   ∩ cin 3559  {csn 4155  {cpr 4157   ↦ cmpt 4683  ⟶wf 5853  –onto→wfo 5855  ‘cfv 5857  ℩crio 6575  (class class class)co 6615  Vtxcvtx 25808  Edgcedg 25873   USGraph cusgr 25971   NeighbVtx cnbgr 26145   FriendGraph cfrgr 27020 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-xnn0 11324  df-z 11338  df-uz 11648  df-fz 12285  df-hash 13074  df-edg 25874  df-upgr 25907  df-umgr 25908  df-usgr 25973  df-nbgr 26149  df-frgr 27021 This theorem is referenced by:  frgrncvvdeqlem8  27071
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