Theorem gropd 29063

Description: If any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property 𝜓, 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) has this property. (Contributed by AV, 11-Oct-2020.)

Hypotheses

Ref	Expression
gropd.g	⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))
gropd.v	⊢ (𝜑 → 𝑉 ∈ 𝑈)
gropd.e	⊢ (𝜑 → 𝐸 ∈ 𝑊)

Assertion

Ref	Expression
gropd	⊢ (𝜑 → [⟨𝑉, 𝐸⟩ / 𝑔]𝜓)

Distinct variable groups:   𝑔,𝐸   𝑔,𝑉   𝜑,𝑔

Allowed substitution hints:   𝜓(𝑔)   𝑈(𝑔)   𝑊(𝑔)

Proof of Theorem gropd

Step	Hyp	Ref	Expression
1		opex 5475	. . 3 ⊢ ⟨𝑉, 𝐸⟩ ∈ V
2	1	a1i 11	. 2 ⊢ (𝜑 → ⟨𝑉, 𝐸⟩ ∈ V)
3		gropd.g	. 2 ⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))
4		gropd.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ 𝑈)
5		gropd.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
6		opvtxfv 29036	. . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
7		opiedgfv 29039	. . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
8	6, 7	jca 511	. . 3 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
9	4, 5, 8	syl2anc 584	. 2 ⊢ (𝜑 → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
10		nfcv 2903	. . 3 ⊢ Ⅎ𝑔⟨𝑉, 𝐸⟩
11		nfv 1912	. . . 4 ⊢ Ⅎ𝑔((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
12		nfsbc1v 3811	. . . 4 ⊢ Ⅎ𝑔[⟨𝑉, 𝐸⟩ / 𝑔]𝜓
13	11, 12	nfim 1894	. . 3 ⊢ Ⅎ𝑔(((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [⟨𝑉, 𝐸⟩ / 𝑔]𝜓)
14		fveqeq2 6916	. . . . 5 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉))
15		fveqeq2 6916	. . . . 5 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
16	14, 15	anbi12d 632	. . . 4 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)))
17		sbceq1a 3802	. . . 4 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → (𝜓 ↔ [⟨𝑉, 𝐸⟩ / 𝑔]𝜓))
18	16, 17	imbi12d 344	. . 3 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → ((((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) ↔ (((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [⟨𝑉, 𝐸⟩ / 𝑔]𝜓)))
19	10, 13, 18	spcgf 3591	. 2 ⊢ (⟨𝑉, 𝐸⟩ ∈ V → (∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) → (((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [⟨𝑉, 𝐸⟩ / 𝑔]𝜓)))
20	2, 3, 9, 19	syl3c 66	1 ⊢ (𝜑 → [⟨𝑉, 𝐸⟩ / 𝑔]𝜓)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2106  Vcvv 3478  [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:  gropeld  29065