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Theorem gropd 27146
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 5363 . . 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 27119 . . . 4 ((𝑉𝑈𝐸𝑊) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
7 opiedgfv 27122 . . . 4 ((𝑉𝑈𝐸𝑊) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
86, 7jca 515 . . 3 ((𝑉𝑈𝐸𝑊) → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
94, 5, 8syl2anc 587 . 2 (𝜑 → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
10 nfcv 2905 . . 3 𝑔𝑉, 𝐸
11 nfv 1922 . . . 4 𝑔((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
12 nfsbc1v 3729 . . . 4 𝑔[𝑉, 𝐸⟩ / 𝑔]𝜓
1311, 12nfim 1904 . . 3 𝑔(((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [𝑉, 𝐸⟩ / 𝑔]𝜓)
14 fveqeq2 6745 . . . . 5 (𝑔 = ⟨𝑉, 𝐸⟩ → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉))
15 fveqeq2 6745 . . . . 5 (𝑔 = ⟨𝑉, 𝐸⟩ → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
1614, 15anbi12d 634 . . . 4 (𝑔 = ⟨𝑉, 𝐸⟩ → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)))
17 sbceq1a 3720 . . . 4 (𝑔 = ⟨𝑉, 𝐸⟩ → (𝜓[𝑉, 𝐸⟩ / 𝑔]𝜓))
1816, 17imbi12d 348 . . 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 3519 . 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 399  wal 1541   = wceq 1543  wcel 2111  Vcvv 3421  [wsbc 3709  cop 4562  cfv 6398  Vtxcvtx 27111  iEdgciedg 27112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337  ax-un 7542
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-sbc 3710  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-br 5069  df-opab 5131  df-mpt 5151  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-iota 6356  df-fun 6400  df-fv 6406  df-1st 7780  df-2nd 7781  df-vtx 27113  df-iedg 27114
This theorem is referenced by:  gropeld  27148
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