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Theorem gropd 29048
Description: If any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property 𝜓, then the ordered pair 𝑉, 𝐸 of the set of vertices and the set of edges (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property. (Contributed by AV, 11-Oct-2020.)
Hypotheses
Ref Expression
gropd.g (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))
gropd.v (𝜑𝑉𝑈)
gropd.e (𝜑𝐸𝑊)
Assertion
Ref Expression
gropd (𝜑[𝑉, 𝐸⟩ / 𝑔]𝜓)
Distinct variable groups:   𝑔,𝐸   𝑔,𝑉   𝜑,𝑔
Allowed substitution hints:   𝜓(𝑔)   𝑈(𝑔)   𝑊(𝑔)

Proof of Theorem gropd
StepHypRef Expression
1 opex 5469 . . 3 𝑉, 𝐸⟩ ∈ V
21a1i 11 . 2 (𝜑 → ⟨𝑉, 𝐸⟩ ∈ V)
3 gropd.g . 2 (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))
4 gropd.v . . 3 (𝜑𝑉𝑈)
5 gropd.e . . 3 (𝜑𝐸𝑊)
6 opvtxfv 29021 . . . 4 ((𝑉𝑈𝐸𝑊) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
7 opiedgfv 29024 . . . 4 ((𝑉𝑈𝐸𝑊) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
86, 7jca 511 . . 3 ((𝑉𝑈𝐸𝑊) → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
94, 5, 8syl2anc 584 . 2 (𝜑 → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
10 nfcv 2905 . . 3 𝑔𝑉, 𝐸
11 nfv 1914 . . . 4 𝑔((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
12 nfsbc1v 3808 . . . 4 𝑔[𝑉, 𝐸⟩ / 𝑔]𝜓
1311, 12nfim 1896 . . 3 𝑔(((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [𝑉, 𝐸⟩ / 𝑔]𝜓)
14 fveqeq2 6915 . . . . 5 (𝑔 = ⟨𝑉, 𝐸⟩ → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉))
15 fveqeq2 6915 . . . . 5 (𝑔 = ⟨𝑉, 𝐸⟩ → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
1614, 15anbi12d 632 . . . 4 (𝑔 = ⟨𝑉, 𝐸⟩ → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)))
17 sbceq1a 3799 . . . 4 (𝑔 = ⟨𝑉, 𝐸⟩ → (𝜓[𝑉, 𝐸⟩ / 𝑔]𝜓))
1816, 17imbi12d 344 . . 3 (𝑔 = ⟨𝑉, 𝐸⟩ → ((((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) ↔ (((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [𝑉, 𝐸⟩ / 𝑔]𝜓)))
1910, 13, 18spcgf 3591 . 2 (⟨𝑉, 𝐸⟩ ∈ V → (∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) → (((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [𝑉, 𝐸⟩ / 𝑔]𝜓)))
202, 3, 9, 19syl3c 66 1 (𝜑[𝑉, 𝐸⟩ / 𝑔]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1538   = wceq 1540  wcel 2108  Vcvv 3480  [wsbc 3788  cop 4632  cfv 6561  Vtxcvtx 29013  iEdgciedg 29014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-2nd 8015  df-vtx 29015  df-iedg 29016
This theorem is referenced by:  gropeld  29050
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