Theorem gropeld 29065

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) → 𝑔 ∈ 𝐶))
gropeld.v	⊢ (𝜑 → 𝑉 ∈ 𝑈)
gropeld.e	⊢ (𝜑 → 𝐸 ∈ 𝑊)

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

Step	Hyp	Ref	Expression
1		gropeld.g	. . 3 ⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔 ∈ 𝐶))
2		gropeld.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ 𝑈)
3		gropeld.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
4	1, 2, 3	gropd 29063	. 2 ⊢ (𝜑 → [⟨𝑉, 𝐸⟩ / 𝑔]𝑔 ∈ 𝐶)
5		sbcel1v 3862	. 2 ⊢ ([⟨𝑉, 𝐸⟩ / 𝑔]𝑔 ∈ 𝐶 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐶)
6	4, 5	sylib 218	1 ⊢ (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2106  [wsbc 3791  ⟨cop 4637  ‘cfv 6563  Vtxcvtx 29028  iEdgciedg 29029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014  df-vtx 29030  df-iedg 29031

This theorem is referenced by:  upgr0eopALT  29148  upgr1eopALT  29149  upgrspanop  29329  umgrspanop  29330  usgrspanop  29331  cplgrop  29469