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Theorem gropeld 29050
Description: If any representation of a graph with vertices 𝑉 and edges 𝐸 is an element of an arbitrary class 𝐶, 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) is an element of this class 𝐶. (Contributed by AV, 11-Oct-2020.)
Hypotheses
Ref Expression
gropeld.g (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔𝐶))
gropeld.v (𝜑𝑉𝑈)
gropeld.e (𝜑𝐸𝑊)
Assertion
Ref Expression
gropeld (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶)
Distinct variable groups:   𝐶,𝑔   𝑔,𝐸   𝑔,𝑉   𝜑,𝑔
Allowed substitution hints:   𝑈(𝑔)   𝑊(𝑔)

Proof of Theorem gropeld
StepHypRef Expression
1 gropeld.g . . 3 (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔𝐶))
2 gropeld.v . . 3 (𝜑𝑉𝑈)
3 gropeld.e . . 3 (𝜑𝐸𝑊)
41, 2, 3gropd 29048 . 2 (𝜑[𝑉, 𝐸⟩ / 𝑔]𝑔𝐶)
5 sbcel1v 3856 . 2 ([𝑉, 𝐸⟩ / 𝑔]𝑔𝐶 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐶)
64, 5sylib 218 1 (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1538   = wceq 1540  wcel 2108  [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:  upgr0eopALT  29133  upgr1eopALT  29134  upgrspanop  29314  umgrspanop  29315  usgrspanop  29316  cplgrop  29454
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