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Theorem gropd 26122
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 (𝜑𝑉𝑈)
gropd.e (𝜑𝐸𝑊)
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
StepHypRef Expression
1 opex 5081 . . 3 𝑉, 𝐸⟩ ∈ V
21a1i 11 . 2 (𝜑 → ⟨𝑉, 𝐸⟩ ∈ V)
3 gropd.g . 2 (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓))
4 gropd.v . . 3 (𝜑𝑉𝑈)
5 gropd.e . . 3 (𝜑𝐸𝑊)
6 opvtxfv 26083 . . . 4 ((𝑉𝑈𝐸𝑊) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
7 opiedgfv 26086 . . . 4 ((𝑉𝑈𝐸𝑊) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
86, 7jca 555 . . 3 ((𝑉𝑈𝐸𝑊) → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
94, 5, 8syl2anc 696 . 2 (𝜑 → ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
10 nfcv 2902 . . 3 𝑔𝑉, 𝐸
11 nfv 1992 . . . 4 𝑔((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
12 nfsbc1v 3596 . . . 4 𝑔[𝑉, 𝐸⟩ / 𝑔]𝜓
1311, 12nfim 1974 . . 3 𝑔(((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [𝑉, 𝐸⟩ / 𝑔]𝜓)
14 fveq2 6352 . . . . . 6 (𝑔 = ⟨𝑉, 𝐸⟩ → (Vtx‘𝑔) = (Vtx‘⟨𝑉, 𝐸⟩))
1514eqeq1d 2762 . . . . 5 (𝑔 = ⟨𝑉, 𝐸⟩ → ((Vtx‘𝑔) = 𝑉 ↔ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉))
16 fveq2 6352 . . . . . 6 (𝑔 = ⟨𝑉, 𝐸⟩ → (iEdg‘𝑔) = (iEdg‘⟨𝑉, 𝐸⟩))
1716eqeq1d 2762 . . . . 5 (𝑔 = ⟨𝑉, 𝐸⟩ → ((iEdg‘𝑔) = 𝐸 ↔ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸))
1815, 17anbi12d 749 . . . 4 (𝑔 = ⟨𝑉, 𝐸⟩ → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) ↔ ((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)))
19 sbceq1a 3587 . . . 4 (𝑔 = ⟨𝑉, 𝐸⟩ → (𝜓[𝑉, 𝐸⟩ / 𝑔]𝜓))
2018, 19imbi12d 333 . . 3 (𝑔 = ⟨𝑉, 𝐸⟩ → ((((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) ↔ (((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [𝑉, 𝐸⟩ / 𝑔]𝜓)))
2110, 13, 20spcgf 3428 . 2 (⟨𝑉, 𝐸⟩ ∈ V → (∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓) → (((Vtx‘⟨𝑉, 𝐸⟩) = 𝑉 ∧ (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) → [𝑉, 𝐸⟩ / 𝑔]𝜓)))
222, 3, 9, 21syl3c 66 1 (𝜑[𝑉, 𝐸⟩ / 𝑔]𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1630   = wceq 1632  wcel 2139  Vcvv 3340  [wsbc 3576  cop 4327  cfv 6049  Vtxcvtx 26073  iEdgciedg 26074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-1st 7333  df-2nd 7334  df-vtx 26075  df-iedg 26076
This theorem is referenced by:  gropeld  26124
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