Theorem gropeld 29604

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 29602	. 2 ⊢ (𝜑 → [⟨𝑉, 𝐸⟩ / 𝑔]𝑔 ∈ 𝐶)
5		sbcel1v 3804	. 2 ⊢ ([⟨𝑉, 𝐸⟩ / 𝑔]𝑔 ∈ 𝐶 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐶)
6	4, 5	sylib 221	1 ⊢ (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  [wsbc 3739  ⟨cop 4590  ‘cfv 6537  Vtxcvtx 29567  iEdgciedg 29568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6493  df-fun 6539  df-fv 6545  df-1st 7999  df-2nd 8000  df-vtx 29569  df-iedg 29570

This theorem is used by:  upgr0eopALT  29687  upgr1eopALT  29688  upgrspanop  29871  umgrspanop  29872  usgrspanop  29873  cplgrop  30011