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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  ontowfo 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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