Theorem gropd 29125

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 5410	. . 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 29098	. . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
7		opiedgfv 29101	. . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
8	6, 7	jca 516	. . 3 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
9	4, 5, 8	syl2anc 590	. 2 ⊢ (𝜑 → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
10		nfcv 2902	. . 3 ⊢ Ⅎ𝑔⟨𝑉, 𝐸⟩
11		nfv 1921	. . . 4 ⊢ Ⅎ𝑔((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
12		nfsbc1v 3750	. . . 4 ⊢ Ⅎ𝑔[⟨𝑉, 𝐸⟩ / 𝑔]𝜓
13	11, 12	nfim 1903	. . 3 ⊢ Ⅎ𝑔(((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [⟨𝑉, 𝐸⟩ / 𝑔]𝜓)
14		fveqeq2 6843	. . . . 5 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉))
15		fveqeq2 6843	. . . . 5 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
16	14, 15	anbi12d 638	. . . 4 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)))
17		sbceq1a 3741	. . . 4 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → (𝜓 ↔ [⟨𝑉, 𝐸⟩ / 𝑔]𝜓))
18	16, 17	imbi12d 345	. . 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 3536	. 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 396  ∀wal 1545   = wceq 1547   ∈ wcel 2119  Vcvv 3432  [wsbc 3730  ⟨cop 4568  ‘cfv 6492  Vtxcvtx 29090  iEdgciedg 29091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-1st 7938  df-2nd 7939  df-vtx 29092  df-iedg 29093

This theorem is referenced by:  gropeld  29127