Theorem gropd 15888

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 4318	. . 3 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → ⟨𝑉, 𝐸⟩ ∈ V)
4	1, 2, 3	syl2anc 411	. 2 ⊢ (𝜑 → ⟨𝑉, 𝐸⟩ ∈ V)
5		gropd.g	. 2 ⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))
6		opvtxfv 15863	. . . 4 ⊢ ((𝑉 ∈ 𝑈 ∧ 𝐸 ∈ 𝑊) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
7		opiedgfv 15866	. . . 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 411	. 2 ⊢ (𝜑 → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
10		nfcv 2372	. . 3 ⊢ Ⅎ𝑔⟨𝑉, 𝐸⟩
11		nfv 1574	. . . 4 ⊢ Ⅎ𝑔((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
12		nfsbc1v 3048	. . . 4 ⊢ Ⅎ𝑔[⟨𝑉, 𝐸⟩ / 𝑔]𝜓
13	11, 12	nfim 1618	. . 3 ⊢ Ⅎ𝑔(((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [⟨𝑉, 𝐸⟩ / 𝑔]𝜓)
14		fveqeq2 5644	. . . . 5 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉))
15		fveqeq2 5644	. . . . 5 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
16	14, 15	anbi12d 473	. . . 4 ⊢ (𝑔 = ⟨𝑉, 𝐸⟩ → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)))
17		sbceq1a 3039	. . . 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 2886	. 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)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1393   = wceq 1395   ∈ wcel 2200  Vcvv 2800  [wsbc 3029  ⟨cop 3670  ‘cfv 5324  Vtxcvtx 15853  iEdgciedg 15854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fo 5330  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-sub 8342  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-9 9199  df-n0 9393  df-dec 9602  df-ndx 13075  df-slot 13076  df-base 13078  df-edgf 15846  df-vtx 15855  df-iedg 15856

This theorem is referenced by:  gropeld  15890