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Theorem gropeld 26745
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 26743 . 2 (𝜑[𝑉, 𝐸⟩ / 𝑔]𝑔𝐶)
5 sbcel1v 3836 . 2 ([𝑉, 𝐸⟩ / 𝑔]𝑔𝐶 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐶)
64, 5sylib 219 1 (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1526   = wceq 1528  wcel 2105  [wsbc 3769  cop 4563  cfv 6348  Vtxcvtx 26708  iEdgciedg 26709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-1st 7678  df-2nd 7679  df-vtx 26710  df-iedg 26711
This theorem is referenced by:  upgr0eopALT  26828  upgr1eopALT  26829  upgrspanop  27006  umgrspanop  27007  usgrspanop  27008  cplgrop  27146
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