Theorem gropeld 27633

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 27631	. 2 ⊢ (𝜑 → [⟨𝑉, 𝐸⟩ / 𝑔]𝑔 ∈ 𝐶)
5		sbcel1v 3797	. 2 ⊢ ([⟨𝑉, 𝐸⟩ / 𝑔]𝑔 ∈ 𝐶 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐶)
6	4, 5	sylib 217	1 ⊢ (𝜑 → ⟨𝑉, 𝐸⟩ ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1538   = wceq 1540   ∈ wcel 2105  [wsbc 3726  ⟨cop 4578  ‘cfv 6473  Vtxcvtx 27596  iEdgciedg 27597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3727  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6425  df-fun 6475  df-fv 6481  df-1st 7891  df-2nd 7892  df-vtx 27598  df-iedg 27599

This theorem is referenced by:  upgr0eopALT  27716  upgr1eopALT  27717  upgrspanop  27894  umgrspanop  27895  usgrspanop  27896  cplgrop  28034