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Theorem gropeld 28866
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 28864 . 2 (𝜑[𝑉, 𝐸⟩ / 𝑔]𝑔𝐶)
5 sbcel1v 3849 . 2 ([𝑉, 𝐸⟩ / 𝑔]𝑔𝐶 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐶)
64, 5sylib 217 1 (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1531   = wceq 1533  wcel 2098  [wsbc 3778  cop 4638  cfv 6553  Vtxcvtx 28829  iEdgciedg 28830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fv 6561  df-1st 7999  df-2nd 8000  df-vtx 28831  df-iedg 28832
This theorem is referenced by:  upgr0eopALT  28949  upgr1eopALT  28950  upgrspanop  29130  umgrspanop  29131  usgrspanop  29132  cplgrop  29270
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