Theorem gropd 16288

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		gropd.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ 𝑈)
2		gropd.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
3		opexg 4368	. . 3 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → ⟨𝑉, 𝐸⟩ ∈ V)
4	1, 2, 3	syl2anc 415	. 2 ⊢ (𝜑 → ⟨𝑉, 𝐸⟩ ∈ V)
5		gropd.g	. 2 ⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))
6		opvtxfv 16263	. . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
7		opiedgfv 16266	. . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
8	6, 7	jca 306	. . 3 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
9	1, 2, 8	syl2anc 415	. 2 ⊢ (𝜑 → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
10		nfcv 2392	. . 3 ⊢ Ⅎ𝑔⟨𝑉, 𝐸⟩
11		nfv 1581	. . . 4 ⊢ Ⅎ𝑔((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
12		nfsbc1v 3070	. . . 4 ⊢ Ⅎ𝑔[⟨𝑉, 𝐸⟩ / 𝑔]𝜓
13	11, 12	nfim 1625	. . 3 ⊢ Ⅎ𝑔(((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [⟨𝑉, 𝐸⟩ / 𝑔]𝜓)
14		fveqeq2 5704	. . . . 5 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉))
15		fveqeq2 5704	. . . . 5 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
16	14, 15	anbi12d 477	. . . 4 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)))
17		sbceq1a 3061	. . . 4 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → (𝜓 ↔ [⟨𝑉, 𝐸⟩ / 𝑔]𝜓))
18	16, 17	imbi12d 234	. . 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 2907	. 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	4, 5, 9, 19	syl3c 63	1 ⊢ (𝜑 → [⟨𝑉, 𝐸⟩ / 𝑔]𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  ∀wal 1400   = wceq 1402   ∈ wcel 2209  Vcvv 2821  [wsbc 3051  ⟨cop 3712  ‘cfv 5377  Vtxcvtx 16253  iEdgciedg 16254

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-sub 8499  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-dec 9778  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256

This theorem is used by:  gropeld  16290