Theorem gropd 26824

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 5321	. . 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 26797	. . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
7		opiedgfv 26800	. . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
8	6, 7	jca 515	. . 3 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
9	4, 5, 8	syl2anc 587	. 2 ⊢ (𝜑 → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
10		nfcv 2955	. . 3 ⊢ Ⅎ𝑔⟨𝑉, 𝐸⟩
11		nfv 1915	. . . 4 ⊢ Ⅎ𝑔((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
12		nfsbc1v 3740	. . . 4 ⊢ Ⅎ𝑔[⟨𝑉, 𝐸⟩ / 𝑔]𝜓
13	11, 12	nfim 1897	. . 3 ⊢ Ⅎ𝑔(((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [⟨𝑉, 𝐸⟩ / 𝑔]𝜓)
14		fveqeq2 6654	. . . . 5 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉))
15		fveqeq2 6654	. . . . 5 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
16	14, 15	anbi12d 633	. . . 4 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)))
17		sbceq1a 3731	. . . 4 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → (𝜓 ↔ [⟨𝑉, 𝐸⟩ / 𝑔]𝜓))
18	16, 17	imbi12d 348	. . 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 3538	. 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 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  Vcvv 3441  [wsbc 3720  ⟨cop 4531  ‘cfv 6324  Vtxcvtx 26789  iEdgciedg 26790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fv 6332  df-1st 7671  df-2nd 7672  df-vtx 26791  df-iedg 26792

This theorem is referenced by:  gropeld  26826