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Theorem grtri 47907
Description: The triangles in a graph. (Contributed by AV, 20-Jul-2025.)
Hypotheses
Ref Expression
grtri.v 𝑉 = (Vtx‘𝐺)
grtri.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
grtri (𝐺𝑊 → (GrTriangles‘𝐺) = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))})
Distinct variable groups:   𝑓,𝐸,𝑡   𝑓,𝐺,𝑡   𝑓,𝑉,𝑡
Allowed substitution hints:   𝑊(𝑡,𝑓)

Proof of Theorem grtri
Dummy variables 𝑒 𝑔 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-grtri 47905 . . 3 GrTriangles = (𝑔 ∈ V ↦ (Vtx‘𝑔) / 𝑣(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))})
21a1i 11 . 2 (𝐺𝑊 → GrTriangles = (𝑔 ∈ V ↦ (Vtx‘𝑔) / 𝑣(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))}))
3 fveq2 6906 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
4 grtri.v . . . . . 6 𝑉 = (Vtx‘𝐺)
53, 4eqtr4di 2795 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
6 fveq2 6906 . . . . . . 7 (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
7 grtri.e . . . . . . 7 𝐸 = (Edg‘𝐺)
86, 7eqtr4di 2795 . . . . . 6 (𝑔 = 𝐺 → (Edg‘𝑔) = 𝐸)
98csbeq1d 3903 . . . . 5 (𝑔 = 𝐺(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))} = 𝐸 / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))})
105, 9csbeq12dv 3908 . . . 4 (𝑔 = 𝐺(Vtx‘𝑔) / 𝑣(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))} = 𝑉 / 𝑣𝐸 / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))})
1110adantl 481 . . 3 ((𝐺𝑊𝑔 = 𝐺) → (Vtx‘𝑔) / 𝑣(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))} = 𝑉 / 𝑣𝐸 / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))})
124fvexi 6920 . . . 4 𝑉 ∈ V
137fvexi 6920 . . . 4 𝐸 ∈ V
14 pweq 4614 . . . . . 6 (𝑣 = 𝑉 → 𝒫 𝑣 = 𝒫 𝑉)
1514adantr 480 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝒫 𝑣 = 𝒫 𝑉)
16 eleq2 2830 . . . . . . . . 9 (𝑒 = 𝐸 → ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ↔ {(𝑓‘0), (𝑓‘1)} ∈ 𝐸))
17 eleq2 2830 . . . . . . . . 9 (𝑒 = 𝐸 → ({(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ↔ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸))
18 eleq2 2830 . . . . . . . . 9 (𝑒 = 𝐸 → ({(𝑓‘1), (𝑓‘2)} ∈ 𝑒 ↔ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))
1916, 17, 183anbi123d 1438 . . . . . . . 8 (𝑒 = 𝐸 → (({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒) ↔ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)))
2019anbi2d 630 . . . . . . 7 (𝑒 = 𝐸 → ((𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒)) ↔ (𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))))
2120exbidv 1921 . . . . . 6 (𝑒 = 𝐸 → (∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒)) ↔ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))))
2221adantl 481 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒)) ↔ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))))
2315, 22rabeqbidv 3455 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → {𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))} = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))})
2412, 13, 23csbie2 3938 . . 3 𝑉 / 𝑣𝐸 / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))} = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))}
2511, 24eqtrdi 2793 . 2 ((𝐺𝑊𝑔 = 𝐺) → (Vtx‘𝑔) / 𝑣(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))} = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))})
26 elex 3501 . 2 (𝐺𝑊𝐺 ∈ V)
274pweqi 4616 . . . . 5 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
28 fvex 6919 . . . . . 6 (Vtx‘𝐺) ∈ V
2928pwex 5380 . . . . 5 𝒫 (Vtx‘𝐺) ∈ V
3027, 29eqeltri 2837 . . . 4 𝒫 𝑉 ∈ V
3130rabex 5339 . . 3 {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))} ∈ V
3231a1i 11 . 2 (𝐺𝑊 → {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))} ∈ V)
332, 25, 26, 32fvmptd 7023 1 (𝐺𝑊 → (GrTriangles‘𝐺) = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {crab 3436  Vcvv 3480  csb 3899  𝒫 cpw 4600  {cpr 4628  cmpt 5225  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156  2c2 12321  3c3 12322  ..^cfzo 13694  Vtxcvtx 29013  Edgcedg 29064  GrTrianglescgrtri 47904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-grtri 47905
This theorem is referenced by:  grtriprop  47908  isgrtri  47910
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