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Theorem gropd 25708
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 4859 . . 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 25681 . . . 4 ((𝑉𝑈𝐸𝑊) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
7 opiedgfv 25684 . . . 4 ((𝑉𝑈𝐸𝑊) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
86, 7jca 553 . . 3 ((𝑉𝑈𝐸𝑊) → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
94, 5, 8syl2anc 691 . 2 (𝜑 → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
10 nfcv 2751 . . 3 𝑔𝑉, 𝐸
11 nfv 1830 . . . 4 𝑔((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
12 nfsbc1v 3422 . . . 4 𝑔[𝑉, 𝐸⟩ / 𝑔]𝜓
1311, 12nfim 1813 . . 3 𝑔(((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [𝑉, 𝐸⟩ / 𝑔]𝜓)
14 fveq2 6103 . . . . . 6 (𝑔 = ⟨𝑉, 𝐸⟩ → (Vtx‘𝑔) = (Vtx‘⟨𝑉, 𝐸⟩))
1514eqeq1d 2612 . . . . 5 (𝑔 = ⟨𝑉, 𝐸⟩ → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉))
16 fveq2 6103 . . . . . 6 (𝑔 = ⟨𝑉, 𝐸⟩ → (iEdg‘𝑔) = (iEdg‘⟨𝑉, 𝐸⟩))
1716eqeq1d 2612 . . . . 5 (𝑔 = ⟨𝑉, 𝐸⟩ → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
1815, 17anbi12d 743 . . . 4 (𝑔 = ⟨𝑉, 𝐸⟩ → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)))
19 sbceq1a 3413 . . . 4 (𝑔 = ⟨𝑉, 𝐸⟩ → (𝜓[𝑉, 𝐸⟩ / 𝑔]𝜓))
2018, 19imbi12d 333 . . 3 (𝑔 = ⟨𝑉, 𝐸⟩ → ((((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) ↔ (((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [𝑉, 𝐸⟩ / 𝑔]𝜓)))
2110, 13, 20spcgf 3261 . 2 (⟨𝑉, 𝐸⟩ ∈ V → (∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) → (((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [𝑉, 𝐸⟩ / 𝑔]𝜓)))
222, 3, 9, 21syl3c 64 1 (𝜑[𝑉, 𝐸⟩ / 𝑔]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1473   = wceq 1475  wcel 1977  Vcvv 3173  [wsbc 3402  cop 4131  cfv 5804  Vtxcvtx 25673  iEdgciedg 25674
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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-2nd 7060  df-vtx 25675  df-iedg 25676
This theorem is referenced by:  gropeld  25710
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